Question

for the following set of data, find the sample standard deviation, to the nearest thousandth. 96, 63, 83, 78, 55, 96, 114, 52

Explanation:

Step1: Calculate the mean

$\bar{x}=\frac{96 + 63+83+78+55+96+114+52}{8}=\frac{637}{8}=79.625$

Step2: Calculate the squared - differences

$(96 - 79.625)^2=(16.375)^2 = 268.140625$
$(63 - 79.625)^2=(-16.625)^2 = 276.390625$
$(83 - 79.625)^2=(3.375)^2 = 11.390625$
$(78 - 79.625)^2=(-1.625)^2 = 2.640625$
$(55 - 79.625)^2=(-24.625)^2 = 606.390625$
$(96 - 79.625)^2=(16.375)^2 = 268.140625$
$(114 - 79.625)^2=(34.375)^2 = 1181.640625$
$(52 - 79.625)^2=(-27.625)^2 = 763.140625$

Step3: Calculate the sum of squared - differences

$S=\sum_{i = 1}^{n}(x_i-\bar{x})^2=268.140625+276.390625 + 11.390625+2.640625+606.390625+268.140625+1181.640625+763.140625=3377.875$

Step4: Calculate the sample variance

$s^2=\frac{S}{n - 1}=\frac{3377.875}{8 - 1}=\frac{3377.875}{7}\approx482.553571$

Step5: Calculate the sample standard deviation

$s=\sqrt{s^2}=\sqrt{482.553571}\approx21.967$

Answer:

$21.967$