Comparing Z and T Statistics: Thick Tails and Sampling Variability

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2024-04-20 23:14

In collections Statistics Demonstrations

This GeoGebra example visualizes and compares the sampling behavior of Z and T statistics, emphasizing how the t distribution’s thick tails create larger variability and wider inference ranges under finite sample sizes. It builds an empirical distribution from repeated random samples and contrasts it with the corresponding Z-statistic behavior. Key concepts include standard deviation and standard error, the construction of Z vs. T statistics, the relationship between sample mean/variance and their sampling distributions, and why heavy tails affect statistical inference. The thick-tail effect is illustrated through the spread of simulated statistic values. Construction highlights: Sequence and Random generate repeated samples (sam), along with derived quantities such as sample means (samMean) and sample standard deviations (samSD). The standard error (samSE) is computed from the simulated dispersion, then Tstatistic and Zstatistic are calculated for comparison. A DotPlot displays the resulting empirical distributions, while interactive buttons and text objects (e.g., button1 and multiple text labels) support explanation and annotation. Suitable for upper-secondary to early undergraduate probability & statistics, especially when teaching sampling distributions, standard error, and hypothesis testing/inference.

GeoGebra Commands & Parameters

Sequence(Sample[pop, n], k, 1, numSam) → sam, {{99.28, 101.38, 106.11, 102.02, 94.32}, {95.66, 99.79, 98.31, 95.5, 102.22}, {101.95, 100.65, 95.29, 99, 94.66}, {95.8, 95.72, 96.89, 102.74, 95.33}, {97.61, 101.92, 95.39, 95.17, 100.94}, {99.74, 97.93, 94.88, 87.62, 95.58}, {104.93, 93.96, 94.88, 101.68, 100.47}, {102.29, 101.61, 98.47, 97.27, 98.44}, {98.44, 106.38, 96.1, 94.63, 96.34}, {107.49, 103.49, 98.92, 91.59, 95.21}, {97.09, 98.89, 103.34, 96.54, 98.99}, {108.03, 102.52, 99.53, 97.56, 101.45}, {89.88, 100.59, 100.74, 98.19, 109}, {99.05, 104.37, 103.27, 103.14, 100.32}, {99.7, 103.03, 103.51, 94.95, 97.79}, {105.66, 104.07, 101.18, 103.46, 99.01}, {96.31, 99.6, 101.82, 101.29, 106.58}, {98.88, 100.17, 96.56, 105.39, 107.91}, {100.26, 96.51, 96.88, 101.88, 96.21}, {101.76, 99.47, 99, 101.44, 100.8}, {106.29, 95.78, 98.87, 97.96, 96.34}, {106.02, 101.88, 100.35, 101.16, 99.98}, {98.69, 100.8, 95.11, 92.56, 94.95}, {98.16, 100.26, 99.62, 98.36, 105.71}, {96.07, 98.98, 97.85, 100.81, 100.86}, {99.74, 100.88, 100.48, 104.05, 107.3}, {110.23, 103.07, 98.73, 100.17, 100.35}, {99.21, 96.21, 103.79, 103.04, 107.29}, {97.42, 101.15, 100.33, 99.25, 104.78}, {95.55, 101.02, 102.27, 95.71, 96.03}, {98.46, 106.27, 100.11, 96.5, 100.42}, {95.92, 103.93, 106.99, 106.9, 95.96}, {94.92, 105.8, 100.12, 100.44, 105.05}, {102.62, 93.7, 105.3, 94.26, 98.67}, {96.36, 105.46, 101.43, 101.45, 92.27}, {100.17, 89.97, 104.16, 98.19, 94.54}, {97.62, 106.73, 98.2, 95.45, 91.03}, {97.39, 102.26, 96.94, 100.9, 96.72}, {102.69, 102.11, 99.18, 92.4, 95.03}, {94.98, 98.16, 105.8, 98.36, 102.49}, {100.81, 91.98, 91.64, 102.94, 96.1}, {101.05, 104.99, 101.87, 100.71, 102.37}, {96.44, 103.59, 102.29, 98.94, 99.76}, {102.3, 104.01, 105.83, 103.43, 97.6}, {98.2, 104.32, 102.5, 99.29, 103.1}, {104.52, 100.8, 96.76, 102.71, 105.67}, {97.86, 99.21, 102.08, 98.69, 101.34}, {96.09, 95.1, 101.06, 98.38, 90.6}, {98.37, 107.76, 97.14, 97.44, 103.54}, {105.13, 101.03, 103.93, 105.79, 96.9}, {100.03, 102.86, 97.94, 99.98, 104.94}, {94.59, 100.26, 99.29, 96.75, 91.43}, {103.51, 98.6, 102.84, 99.65, 99.12}, {96.51, 97.97, 99.13, 96.53, 89.96}, {100.33, 93.04, 105.08, 99.87, 97.19}, {101.01, 103.09, 101.21, 101.39, 101.06}, {101.51, 94.69, 101.12, 98.34, 96.65}, {98.52, 104.47, 89.61, 95.73, 105.01}, {97.86, 102.16, 96.98, 102.28, 105.3}, {98.07, 101.27, 96.18, 98.67, 102.67}, {102.49, 93.26, 98.66, 104.07, 100.81}, {99.13, 104.27, 99.25, 99.06, 104.67}, {107, 98.15, 107.57, 95.97, 94.28}, {92.35, 94.42, 93.47, 104.16, 97.17}, {100.65, 98.1, 104.56, 98.46, 98.87}, {104.04, 100.1, 104.07, 104.55, 92.71}, {98.64, 104.24, 99.64, 98.08, 104.73}, {97.18, 97.45, 104.26, 94.58, 107.05}, {99.82, 95.7, 102.57, 97.72, 104.69}, {106.54, 97.9, 96.1, 97.37, 102.93}, {102.53, 93.17, 102.25, 95.83, 99.13}, {100.68, 98.22, 106.78, 97.93, 99.08}, {103.74, 102.28, 97.92, 99.82, 105.45}, {101.84, 97.85, 102.23, 96.17, 98.57}, {104.14, 96.98, 106.5, 97.78, 87.72}, {109.52, 95.7, 108.75, 94.91, 102.37}, {99.91, 100.67, 99.45, 97.88, 99.94}, {98.41, 102.27, 98.11, 98.27, 97.7}, {95.23, 96.45, 97.94, 98.64, 99.97}, {97.76, 96.36, 100.7, 98.54, 99.69}, {96.4, 104.31, 101.19, 103.36, 104.02}, {101.21, 107.25, 103.35, 94.54, 103.72}, {91.46, 102.99, 99.33, 100.57, 108.66}, {99.54, 100.43, 94.69, 100.68, 104.58}, {96.58, 96.95, 106.58, 109.01, 98.07}, {101.45, 99.17, 102.74, 99.94, 99.79}, {99.13, 96.19, 96.18, 106.19, 89.42}, {98.31, 103.93, 98.95, 100.33, 98.75}, {95.29, 104.32, 96.86, 94.39, 97.16}, {103.24, 97.33, 103.23, 102.93, 98.58}, {104.59, 99.29, 106.49, 95.34, 97.12}, {93.4, 100.16, 95.27, 92.55, 99.79}, {96.07, 103.98, 99.24, 101.84, 101.83}, {102.99, 105.31, 99.48, 103.51, 90.47}, {99.56, 96.12, 96.36, 105.91, 99.9}, {94.56, 93.78, 98.43, 96.34, 96.51}, {98.02, 103.02, 95.25, 103.39, 97.3}, {97.4, 99.71, 99.87, 103.08, 103.91}, {104.02, 94.6, 99.26, 97.39, 94.09}, {96.19, 98.22, 101.08, 96.16, 98.54}, {95.05, 105.01, 108.49, 93.35, 98.78}, {106.59, 98.58, 97.08, 102.3, 101.46}, {93.52, 100.03, 104.92, 99.33, 102.27}, {102.43, 98.22, 102.42, 100.97, 99.01}, {101.65, 96.9, 100.07, 99.81, 97.43}, {97.98, 98.79, 98.09, 95.96, 97.8}, {102.75, 96.53, 103.01, 102.65, 100.86}, {100.51, 106.98, 97.77, 100.79, 93.93}, {96.13, 97.72, 97.2, 99.55, 102.66}, {102.84, 101.88, 109.77, 97.15, 101.53}, {99.12, 104.02, 89.05, 100.01, 102.37}, {100.26, 101.32, 98.57, 106.21, 96.1}, {106.15, 104.45, 94.17, 103.36, 107.01}, {96, 105.15, 102.95, 106.01, 101.32}, {101.03, 100.17, 101.69, 97.49, 106.41}, {96.89, 103.69, 100.51, 101.02, 104.54}, {100.92, 106.29, 104.36, 103.83, 101.2}, {94.3, 105.3, 95.72, 100.92, 95.6}, {101.65, 96.15, 103.3, 98.01, 98.74}, {98.67, 99.88, 98.34, 100.09, 97.07}, {94.26, 95.18, 94.43, 102.01, 96.66}, {97.06, 99.82, 98.89, 103.69, 98.36}, {96.4, 95.58, 104.22, 97.45, 96.8}, {102.42, 94.64, 87.05, 110.77, 97.75}, {103.96, 95.05, 101.88, 97.2, 96.13}, {107.17, 94.03, 99.82, 95.63, 105.32}, {104.36, 102.25, 103.82, 98.8, 97.65}, {99.26, 102.24, 104.65, 95.94, 95.42}, {95.92, 100.38, 104.11, 102.44, 100.2}, {99.05, 106.7, 106.33, 108.49, 97.94}, {104.99, 97.17, 99.06, 105.02, 95.34}, {104.75, 106.27, 98.58, 97.04, 95.21}, {99.65, 96.56, 97.01, 98.27, 112.32}, {100.5, 106.58, 106.26, 87.75, 103.32}, {108.2, 97.53, 104.65, 99.47, 106.06}, {106.7, 101.02, 97.01, 100.35, 97.76}, {99.46, 103.11, 99.37, 96.44, 96.06}, {100.08, 107.87, 100.57, 99.32, 107.11}, {97.66, 103.74, 104.98, 104.76, 101.12}, {101.57, 105.91, 102.37, 100.03, 101.45}, {110.06, 101.68, 99.56, 105, 105.18}, {100.35, 100.24, 97.67, 99.23, 92.19}, {99.09, 94.31, 98.1, 101.76, 96.97}, {100.05, 102.05, 101.54, 101.03, 106.36}, {105.37, 102.34, 98.53, 105.73, 99.41}, {87.62, 98.31, 96.88, 99.76, 101.45}, {95.83, 97.88, 101.61, 102.88, 103.83}, {98.99, 104.7, 102.16, 100.05, 106.15}, {97.93, 100.78, 98.36, 105.55, 100.03}, {107.33, 101.48, 100.76, 95.25, 102.22}, {97.95, 95.32, 101.7, 101.4, 100.24}, {107.11, 96.43, 100.75, 99.76, 98}, {103.24, 100.35, 101.59, 103.73, 96.12}, {99.41, 99.38, 101.72, 91.46, 106.36}, {94.09, 100.68, 102.9, 100.62, 99.71}, {101.89, 100.16, 97.28, 97.25, 94.89}, {105.32, 96.38, 107.58, 97.78, 98.84}, {102.61, 98.63, 100.59, 99.04, 103.73}, {95.93, 105.93, 95.74, 98.78, 97.48}, {101.27, 97.32, 97.62, 94.57, 97.96}, {105.46, 100.08, 100.09, 104.93, 95.09}, {108.66, 102.16, 107.91, 99.48, 97.06}, {100.2, 102.17, 102.34, 98.66, 100.27}, {104.27, 98.25, 99.96, 97.92, 102.73}, {100.57, 103.36, 103.16, 102.29, 97.06}, {111.01, 99.34, 95.26, 93.23, 88.47}, {99.85, 99.87, 95.99, 102.49, 104.18}, {103.72, 97.39, 96.91, 97.34, 99.27}, {96.06, 95.17, 93.85, 101.73, 91.98}, {102.42, 96.03, 103.52, 97.18, 96.25}, {102.91, 98.16, 99.51, 106.25, 98.21}, {95.04, 101.3, 100.26, 102.15, 98.25}, {100.08, 103.47, 99.48, 102.95, 104.03}, {97.17, 97.53, 101.84, 100.8, 101.82}, {101.95, 103.56, 101.06, 105.75, 103.46}, {96.65, 100.01, 97.77, 97.28, 99.92}, {103.37, 101.98, 97.32, 97.62, 101.72}, {97.13, 104.77, 98.43, 94.58, 95.48}, {99.87, 100.77, 99.69, 100.94, 95.58}, {100.62, 99.95, 95.17, 104.51, 95.26}, {94.44, 98.56, 102.88, 100.85, 106.73}, {99.05, 92.99, 103.34, 94.17, 96.47}, {96.65, 98.06, 101.21, 99.17, 100.43}, {98.28, 95.57, 98.31, 99.06, 103.48}, {94.13, 99.87, 101.83, 101.92, 101.14}, {97.29, 99.56, 106.29, 96.46, 101.26}, {104.04, 98.77, 99.18, 95.92, 103.33}, {94.48, 105.77, 101.24, 100.85, 101.72}, {96.96, 102.66, 100.13, 101.15, 102.34}, {100.31, 99.58, 101.78, 103.78, 100.15}, {98.5, 90.07, 102.99, 100.12, 105.18}, {97.23, 100.43, 99.09, 95.49, 102.49}, {98.77, 95.09, 106.25, 97.63, 98.35}, {106.98, 97.33, 91.04, 101.06, 106.34}, {105.99, 106.45, 96.25, 92.06, 98.9}, {100.48, 99.94, 109.42, 106.59, 96.77}, {100.1, 102.15, 101.72, 96.93, 99.8}, {94.28, 106.66, 94.64, 105.5, 104.65}, {98.42, 98.44, 95.44, 100.29, 99.27}, {102.43, 98.47, 101.02, 98.04, 97.69}}
Sequence(mean[Element[sam, k]], k, 1, Length[sam]) → samMean
Sequence(stdev[Element[sam, k]], k, 1, Length[sam]) → samSD
DotPlot(Tstatistic, true) → l1
DotPlot(Zstatistic, true) → l2
Random(1, 2) → a
Random(1, 2) → b
Random(1, 2) → c
Random(1, 2) → d
Random(1, 2) → e
Random(1, 2) → f
Random(1, 2) → g
Random(1, 2) → h
Random(1, 2) → i
Random(1, 2) → j
Random(1, 2) → k
Random(1, 2) → l
Random(1, 2) → o
Random(1, 2) → p
Random(1, 2) → q
Random(1, 2) → r
Random(1, 2) → s

AI Drawing Prompt Reference

First, generate multiple random samples, each with five data points. Then, compute the mean and standard deviation for each sample. Next, calculate the T-statistic and Z-statistic based on these samples and plot two dot plots: one showing the distribution of T-statistics and another showing the distribution of Z-statistics. Finally, compare the two plots to illustrate that under finite sample sizes, the t-distribution's heavy tails lead to a wider spread for T-statistics, while Z-statistics are more concentrated.

The prompt above was compiled from the original drawing commands and is for reference only.