Question

1. find the standard deviation *

here are the foot lengths (in centimeters) for a random sample of seven 14 - year - olds from the united kingdom.
25 22 20 25 24 24 28
a. sd = 2.0
b. sd = 1.87
c. sd = 2.52
d. sd = 2.13

Explanation:

Step1: Calculate the mean

The data set is \(25,22,20,25,24,24,28\).
The mean \(\bar{x}=\frac{25 + 22+20+25+24+24+28}{7}=\frac{168}{7} = 24\).

Step2: Calculate the squared differences

\((25 - 24)^2=1\), \((22 - 24)^2 = 4\), \((20 - 24)^2=16\), \((25 - 24)^2=1\), \((24 - 24)^2=0\), \((24 - 24)^2=0\), \((28 - 24)^2=16\).

Step3: Calculate the variance

The variance \(s^{2}=\frac{1+4 + 16+1+0+0+16}{7}=\frac{38}{7}\approx5.43\).

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{\frac{38}{7}}\approx2.33\) (using a calculator for the square - root). Wait, maybe there was a miscalculation. Let's re - calculate:

Re - calculation of variance and standard deviation

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\) (since it's a sample).
\(\sum_{i=1}^{7}(x_{i}-\bar{x})^{2}=(25 - 24)^2+(22 - 24)^2+(20 - 24)^2+(25 - 24)^2+(24 - 24)^2+(24 - 24)^2+(28 - 24)^2\)
\(=1 + 4+16 + 1+0+0+16=38\)
\(n = 7\), so \(s=\sqrt{\frac{38}{7-1}}=\sqrt{\frac{38}{6}}\approx\sqrt{6.33}\approx2.52\)

Answer:

C. \(Sd = 2.52\)