id int64 1 14M | domain stringclasses 6
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values | title stringlengths 14 86 | content stringlengths 203 553 | key_equations stringclasses 23
values | prerequisites stringclasses 29
values | learning_objective stringclasses 37
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5,101 | mathematics | logarithms | common_logarithm | 3 | worked_example | Common logarithm of 179.3 | The common logarithm log₁₀(179.3) = 2.254. Logarithms convert multiplicative relationships into additive ones and are indispensable for expressing quantities that span many orders of magnitude (pH, sound intensity, stellar magnitudes, earthquake energy). | log10(x) | exponents | Compute and interpret common logarithms of scientific quantities. |
5,102 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 4.554 t² at t = 5.127 | Position given by s(t) = 4.554 t². The instantaneous velocity is the derivative ds/dt = 2 4.554 t = 46.69 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,103 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.473 t² at t = 6.264 | Position given by s(t) = 1.473 t². The instantaneous velocity is the derivative ds/dt = 2 1.473 t = 18.45 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,104 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.938 t² at t = 1.464 | Position given by s(t) = 1.938 t². The instantaneous velocity is the derivative ds/dt = 2 1.938 t = 5.672 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,105 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.234 t² at t = 3.122 | Position given by s(t) = 3.234 t². The instantaneous velocity is the derivative ds/dt = 2 3.234 t = 20.2 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,106 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.91 t² at t = 9.274 | Position given by s(t) = 1.91 t². The instantaneous velocity is the derivative ds/dt = 2 1.91 t = 35.43 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,107 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 0.6441 t² at t = 1.776 | Position given by s(t) = 0.6441 t². The instantaneous velocity is the derivative ds/dt = 2 0.6441 t = 2.287 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,108 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.774 t² at t = 3.583 | Position given by s(t) = 3.774 t². The instantaneous velocity is the derivative ds/dt = 2 3.774 t = 27.05 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,109 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.899 t² at t = 4.431 | Position given by s(t) = 3.899 t². The instantaneous velocity is the derivative ds/dt = 2 3.899 t = 34.55 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,110 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 0.8437 t² at t = 1.929 | Position given by s(t) = 0.8437 t². The instantaneous velocity is the derivative ds/dt = 2 0.8437 t = 3.255 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,111 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.564 t² at t = 2.4 | Position given by s(t) = 1.564 t². The instantaneous velocity is the derivative ds/dt = 2 1.564 t = 7.509 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,112 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 0.5312 t² at t = 2.83 | Position given by s(t) = 0.5312 t². The instantaneous velocity is the derivative ds/dt = 2 0.5312 t = 3.006 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,113 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 2.906 t² at t = 7.746 | Position given by s(t) = 2.906 t². The instantaneous velocity is the derivative ds/dt = 2 2.906 t = 45.03 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,114 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.378 t² at t = 5.545 | Position given by s(t) = 1.378 t². The instantaneous velocity is the derivative ds/dt = 2 1.378 t = 15.28 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,115 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.528 t² at t = 0.7076 | Position given by s(t) = 1.528 t². The instantaneous velocity is the derivative ds/dt = 2 1.528 t = 2.162 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,116 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.883 t² at t = 1.146 | Position given by s(t) = 3.883 t². The instantaneous velocity is the derivative ds/dt = 2 3.883 t = 8.901 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,117 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.951 t² at t = 7.258 | Position given by s(t) = 3.951 t². The instantaneous velocity is the derivative ds/dt = 2 3.951 t = 57.35 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,118 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.638 t² at t = 8.608 | Position given by s(t) = 1.638 t². The instantaneous velocity is the derivative ds/dt = 2 1.638 t = 28.21 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,119 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 4.535 t² at t = 4.735 | Position given by s(t) = 4.535 t². The instantaneous velocity is the derivative ds/dt = 2 4.535 t = 42.95 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,120 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 4.434 t² at t = 2.249 | Position given by s(t) = 4.434 t². The instantaneous velocity is the derivative ds/dt = 2 4.434 t = 19.95 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,121 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 2.395 t² at t = 4.878 | Position given by s(t) = 2.395 t². The instantaneous velocity is the derivative ds/dt = 2 2.395 t = 23.36 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,122 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.732 t² at t = 6.938 | Position given by s(t) = 1.732 t². The instantaneous velocity is the derivative ds/dt = 2 1.732 t = 24.03 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,123 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.55 t² at t = 2.926 | Position given by s(t) = 3.55 t². The instantaneous velocity is the derivative ds/dt = 2 3.55 t = 20.77 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,124 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.763 t² at t = 0.7198 | Position given by s(t) = 3.763 t². The instantaneous velocity is the derivative ds/dt = 2 3.763 t = 5.418 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,125 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 0.684 t² at t = 4.402 | Position given by s(t) = 0.684 t². The instantaneous velocity is the derivative ds/dt = 2 0.684 t = 6.023 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,126 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 2.042 t² at t = 5.12 | Position given by s(t) = 2.042 t². The instantaneous velocity is the derivative ds/dt = 2 2.042 t = 20.91 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,127 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 0.3354 t² at t = 5.52 | Position given by s(t) = 0.3354 t². The instantaneous velocity is the derivative ds/dt = 2 0.3354 t = 3.702 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,128 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 4.574 t² at t = 5.366 | Position given by s(t) = 4.574 t². The instantaneous velocity is the derivative ds/dt = 2 4.574 t = 49.09 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,129 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 0.1794 t² at t = 4.653 | Position given by s(t) = 0.1794 t². The instantaneous velocity is the derivative ds/dt = 2 0.1794 t = 1.67 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,130 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 0.2572 t² at t = 5.29 | Position given by s(t) = 0.2572 t². The instantaneous velocity is the derivative ds/dt = 2 0.2572 t = 2.721 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,131 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 4.012 t² at t = 8.006 | Position given by s(t) = 4.012 t². The instantaneous velocity is the derivative ds/dt = 2 4.012 t = 64.24 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,132 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 2.511 t² at t = 5.961 | Position given by s(t) = 2.511 t². The instantaneous velocity is the derivative ds/dt = 2 2.511 t = 29.93 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,133 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.208 t² at t = 7.477 | Position given by s(t) = 3.208 t². The instantaneous velocity is the derivative ds/dt = 2 3.208 t = 47.98 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,134 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 2.87 t² at t = 9.755 | Position given by s(t) = 2.87 t². The instantaneous velocity is the derivative ds/dt = 2 2.87 t = 55.99 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,135 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.925 t² at t = 5.552 | Position given by s(t) = 3.925 t². The instantaneous velocity is the derivative ds/dt = 2 3.925 t = 43.58 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,136 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 4.013 t² at t = 8.54 | Position given by s(t) = 4.013 t². The instantaneous velocity is the derivative ds/dt = 2 4.013 t = 68.54 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,137 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 0.5299 t² at t = 9.264 | Position given by s(t) = 0.5299 t². The instantaneous velocity is the derivative ds/dt = 2 0.5299 t = 9.817 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,138 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.089 t² at t = 9.933 | Position given by s(t) = 3.089 t². The instantaneous velocity is the derivative ds/dt = 2 3.089 t = 61.37 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,139 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.784 t² at t = 7.788 | Position given by s(t) = 3.784 t². The instantaneous velocity is the derivative ds/dt = 2 3.784 t = 58.94 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,140 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 2.212 t² at t = 3.13 | Position given by s(t) = 2.212 t². The instantaneous velocity is the derivative ds/dt = 2 2.212 t = 13.85 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,141 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 2.311 t² at t = 8.865 | Position given by s(t) = 2.311 t². The instantaneous velocity is the derivative ds/dt = 2 2.311 t = 40.98 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,142 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.721 t² at t = 6.679 | Position given by s(t) = 3.721 t². The instantaneous velocity is the derivative ds/dt = 2 3.721 t = 49.7 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,143 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 0.4578 t² at t = 6.654 | Position given by s(t) = 0.4578 t². The instantaneous velocity is the derivative ds/dt = 2 0.4578 t = 6.093 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,144 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.765 t² at t = 5.513 | Position given by s(t) = 1.765 t². The instantaneous velocity is the derivative ds/dt = 2 1.765 t = 19.46 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,145 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.494 t² at t = 8.292 | Position given by s(t) = 3.494 t². The instantaneous velocity is the derivative ds/dt = 2 3.494 t = 57.94 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,146 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 2.219 t² at t = 0.7041 | Position given by s(t) = 2.219 t². The instantaneous velocity is the derivative ds/dt = 2 2.219 t = 3.124 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,147 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 2.916 t² at t = 0.5604 | Position given by s(t) = 2.916 t². The instantaneous velocity is the derivative ds/dt = 2 2.916 t = 3.268 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,148 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 4.158 t² at t = 0.6793 | Position given by s(t) = 4.158 t². The instantaneous velocity is the derivative ds/dt = 2 4.158 t = 5.649 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,149 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.204 t² at t = 4.558 | Position given by s(t) = 1.204 t². The instantaneous velocity is the derivative ds/dt = 2 1.204 t = 10.97 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,150 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.368 t² at t = 2.617 | Position given by s(t) = 1.368 t². The instantaneous velocity is the derivative ds/dt = 2 1.368 t = 7.163 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,151 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.405 t² at t = 3.603 | Position given by s(t) = 3.405 t². The instantaneous velocity is the derivative ds/dt = 2 3.405 t = 24.54 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,152 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 2.343 t² at t = 6.821 | Position given by s(t) = 2.343 t². The instantaneous velocity is the derivative ds/dt = 2 2.343 t = 31.96 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,153 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.873 t² at t = 5.957 | Position given by s(t) = 3.873 t². The instantaneous velocity is the derivative ds/dt = 2 3.873 t = 46.14 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,154 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.838 t² at t = 8.863 | Position given by s(t) = 1.838 t². The instantaneous velocity is the derivative ds/dt = 2 1.838 t = 32.58 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,155 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 4.349 t² at t = 1.95 | Position given by s(t) = 4.349 t². The instantaneous velocity is the derivative ds/dt = 2 4.349 t = 16.96 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,156 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.888 t² at t = 2.458 | Position given by s(t) = 3.888 t². The instantaneous velocity is the derivative ds/dt = 2 3.888 t = 19.11 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,157 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 0.9898 t² at t = 0.7491 | Position given by s(t) = 0.9898 t². The instantaneous velocity is the derivative ds/dt = 2 0.9898 t = 1.483 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,158 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.049 t² at t = 8.525 | Position given by s(t) = 3.049 t². The instantaneous velocity is the derivative ds/dt = 2 3.049 t = 51.99 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,159 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 3.919 t² at t = 5.631 | Position given by s(t) = 3.919 t². The instantaneous velocity is the derivative ds/dt = 2 3.919 t = 44.14 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,160 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 4.778 t² at t = 3.611 | Position given by s(t) = 4.778 t². The instantaneous velocity is the derivative ds/dt = 2 4.778 t = 34.5 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,161 | mathematics | calculus | derivative_as_rate | 6 | worked_example | Instantaneous rate of change of s = 1.645 t² at t = 2.686 | Position given by s(t) = 1.645 t². The instantaneous velocity is the derivative ds/dt = 2 1.645 t = 8.838 (in consistent units). The derivative quantifies the local rate of change and is foundational to continuous models in physics, chemistry, and biology. | ds/dt = lim (Δs/Δt) | linear_relation | Interpret the derivative as an instantaneous rate of change. |
5,162 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 8 observations | Given the sample values ['28.15', '24.7', '99.19', '30.97', '92.79', '79.51', '12.46', '88.99'], the sample mean is x̄ = 57.09 and the sample standard deviation is s = 36.11. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estim... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,163 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['21.08', '95.32', '12.29', '88.08', '86.88', '28.77', '14.43', '49.47', '27.88', '79.39', '63.12', '21.22'], the sample mean is x̄ = 48.99 and the sample standard deviation is s = 31.88. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean a... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,164 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 7 observations | Given the sample values ['55.6', '62.05', '56.48', '24.66', '28.87', '94.66', '76.66'], the sample mean is x̄ = 57 and the sample standard deviation is s = 24.73. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in meas... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,165 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 10 observations | Given the sample values ['77.47', '38.52', '27.19', '98.66', '81.98', '42.39', '32.98', '20.08', '51.13', '10.65'], the sample mean is x̄ = 48.1 and the sample standard deviation is s = 28.97. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,166 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 11 observations | Given the sample values ['46.03', '87.17', '41.92', '95.12', '85.29', '49.07', '86.3', '37.06', '99.97', '17.6', '67.92'], the sample mean is x̄ = 64.86 and the sample standard deviation is s = 27.69. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the b... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,167 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 11 observations | Given the sample values ['88.83', '11.43', '96.78', '75.67', '90.76', '14.08', '86.27', '25.96', '10.83', '56.68', '22.62'], the sample mean is x̄ = 52.72 and the sample standard deviation is s = 35.96. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,168 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['78.61', '15.51', '82.21', '87.97', '69.49', '43.45', '47.34', '24.17', '25.52', '94.27', '96.88', '76.78'], the sample mean is x̄ = 61.85 and the sample standard deviation is s = 29.19. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean a... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,169 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 7 observations | Given the sample values ['42.86', '93.25', '80.79', '12.7', '28.37', '46.33', '43.59'], the sample mean is x̄ = 49.7 and the sample standard deviation is s = 28.23. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in me... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,170 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 9 observations | Given the sample values ['65.97', '12.56', '12.32', '12.56', '52.35', '76.15', '22.99', '46.69', '58.55'], the sample mean is x̄ = 40.02 and the sample standard deviation is s = 25.21. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncerta... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,171 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 7 observations | Given the sample values ['24.24', '11.34', '90.75', '18.29', '98.94', '20.12', '52.81'], the sample mean is x̄ = 45.21 and the sample standard deviation is s = 36.42. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in ... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,172 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 11 observations | Given the sample values ['87.05', '58.31', '22.1', '64.29', '67.32', '38.03', '52.25', '91.26', '21.54', '69.07', '42.9'], the sample mean is x̄ = 55.83 and the sample standard deviation is s = 23.29. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the b... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,173 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 6 observations | Given the sample values ['53', '77.38', '86.68', '27.67', '29.74', '21.68'], the sample mean is x̄ = 49.36 and the sample standard deviation is s = 27.61. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,174 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 7 observations | Given the sample values ['30.3', '17.4', '18.53', '46.35', '89.82', '19.47', '30.36'], the sample mean is x̄ = 36.03 and the sample standard deviation is s = 25.8. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in mea... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,175 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 7 observations | Given the sample values ['75.7', '37.1', '73.94', '51.59', '63.17', '56.52', '98.89'], the sample mean is x̄ = 65.27 and the sample standard deviation is s = 19.9. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in mea... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,176 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 9 observations | Given the sample values ['78.39', '21.85', '47.98', '69.51', '62.14', '94.12', '13.08', '91.54', '44.65'], the sample mean is x̄ = 58.14 and the sample standard deviation is s = 28.72. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncerta... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,177 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 5 observations | Given the sample values ['81.48', '98.16', '11.29', '67.12', '85.18'], the sample mean is x̄ = 68.65 and the sample standard deviation is s = 33.92. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,178 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 8 observations | Given the sample values ['24.57', '39.03', '61.49', '53.55', '40.95', '66.58', '83.53', '61.95'], the sample mean is x̄ = 53.96 and the sample standard deviation is s = 18.56. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty esti... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,179 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['99.34', '50.79', '33.65', '49.59', '77.05', '68.99', '56.3', '40.36', '99.7', '26.77', '40.69', '88.48'], the sample mean is x̄ = 60.98 and the sample standard deviation is s = 25.33. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,180 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['69.69', '70.12', '40.81', '56.72', '15.68', '42.89', '67.81', '96.83', '21.36', '30.44', '51.17', '20'], the sample mean is x̄ = 48.63 and the sample standard deviation is s = 24.76. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and ... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,181 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 9 observations | Given the sample values ['21.37', '41.29', '50.73', '59.8', '94.56', '96.97', '97.45', '26.44', '83.8'], the sample mean is x̄ = 63.6 and the sample standard deviation is s = 30.55. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertaint... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,182 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 8 observations | Given the sample values ['22.75', '71.25', '79.83', '27.88', '77.01', '57.3', '80.19', '60.66'], the sample mean is x̄ = 59.61 and the sample standard deviation is s = 22.81. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estim... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,183 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 5 observations | Given the sample values ['82.02', '42.64', '30.74', '95.24', '57.78'], the sample mean is x̄ = 61.68 and the sample standard deviation is s = 26.81. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measurement. | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,184 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 10 observations | Given the sample values ['24.29', '71.24', '67.14', '87.3', '30.36', '93.76', '74.45', '58.67', '95.96', '48.16'], the sample mean is x̄ = 65.13 and the sample standard deviation is s = 24.94. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,185 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 8 observations | Given the sample values ['35.06', '96.32', '95.4', '36.68', '91.09', '97.48', '90.44', '52.12'], the sample mean is x̄ = 74.32 and the sample standard deviation is s = 27.92. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estim... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,186 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 7 observations | Given the sample values ['74.94', '37', '16.74', '73.75', '70.13', '28.79', '76.01'], the sample mean is x̄ = 53.91 and the sample standard deviation is s = 25.45. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in mea... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,187 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 6 observations | Given the sample values ['15.33', '46.93', '22.94', '84.02', '78.81', '82.63'], the sample mean is x̄ = 55.11 and the sample standard deviation is s = 31.11. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measureme... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,188 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 8 observations | Given the sample values ['45.21', '24.35', '47.71', '63.23', '32.37', '41.54', '30.37', '95.41'], the sample mean is x̄ = 47.52 and the sample standard deviation is s = 22.81. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty esti... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,189 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 7 observations | Given the sample values ['90.34', '21.11', '76.57', '94.99', '23.67', '49.9', '70.5'], the sample mean is x̄ = 61.01 and the sample standard deviation is s = 30.16. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in me... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,190 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 10 observations | Given the sample values ['10.97', '20.56', '38', '19.24', '66.57', '52.4', '69.44', '36.89', '21.24', '76.86'], the sample mean is x̄ = 41.22 and the sample standard deviation is s = 23.78. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for un... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,191 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['23.83', '89.22', '77.88', '74.8', '88.77', '95.45', '13.69', '85.42', '32.99', '66.42', '48.33', '25.16'], the sample mean is x̄ = 60.16 and the sample standard deviation is s = 29.69. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean an... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,192 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['78.26', '30.22', '19.27', '99.78', '64.3', '63.91', '42.63', '88.24', '69.84', '85.3', '96.69', '47.95'], the sample mean is x̄ = 65.53 and the sample standard deviation is s = 26.04. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,193 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 6 observations | Given the sample values ['87.87', '35.22', '75.16', '34.83', '32.65', '55.7'], the sample mean is x̄ = 53.57 and the sample standard deviation is s = 23.55. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimates in measuremen... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,194 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 11 observations | Given the sample values ['70.03', '32.61', '42.1', '11.16', '74.18', '53.7', '62.72', '29.92', '49.04', '33.92', '99.76'], the sample mean is x̄ = 50.83 and the sample standard deviation is s = 24.83. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the b... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,195 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['60.46', '85.37', '43.82', '56.71', '93.27', '44.43', '64.15', '63.13', '81.42', '95.87', '61.81', '20.57'], the sample mean is x̄ = 64.25 and the sample standard deviation is s = 22.11. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean a... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,196 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 8 observations | Given the sample values ['42.46', '70.53', '91.34', '47.96', '45.44', '76.99', '41.26', '80.93'], the sample mean is x̄ = 62.11 and the sample standard deviation is s = 20. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertainty estimat... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,197 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['61.6', '57.05', '94.62', '83.18', '66.53', '79.99', '70.22', '12.31', '61.02', '12.17', '95.14', '18.89'], the sample mean is x̄ = 59.39 and the sample standard deviation is s = 29.81. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean an... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,198 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 11 observations | Given the sample values ['82.17', '75.62', '86.45', '32.82', '83.65', '40.96', '97.62', '32.38', '50.14', '14.3', '51.24'], the sample mean is x̄ = 58.85 and the sample standard deviation is s = 27.43. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the ... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,199 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 9 observations | Given the sample values ['79.18', '24.45', '60.26', '87.71', '68.35', '76.66', '77.8', '80.29', '25.49'], the sample mean is x̄ = 64.47 and the sample standard deviation is s = 23.68. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean and is the basis for uncertai... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
5,200 | mathematics | statistics | descriptive_stats | 4 | worked_example | Sample mean and standard deviation of 12 observations | Given the sample values ['87.1', '40.04', '65.48', '76.59', '85.26', '30.48', '84.58', '19.12', '74.97', '67.55', '38.16', '54.75'], the sample mean is x̄ = 60.34 and the sample standard deviation is s = 23.37. The mean locates the center of the data; the standard deviation quantifies typical deviation from the mean an... | x̄ = (1/n) Σ x_i; s² = 1/(n-1) Σ (x_i - x̄)² | basic arithmetic | Compute and interpret the sample mean and sample standard deviation. |
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