Question

for the following list of data values: 32, 20, 23, 26, 24. a. find the average. answer: 25. b. find the sd. answer: blank. c. convert each value to standard units using the average and sd of the list. i. 32, z = 1.57; ii. 20, z = -1.12; iii. 23, z = -0.45; iv. 26, z = 0.22; v. 24, z = -0.22. d. find the average of the converted list. avg = 0. e. find the sd of the converted list. sd = 1.

Explanation:

Step1: Calculate the average of the original data list

The data list is \(32, 20, 23, 26, 24\). The formula for the average \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 5\), \(\sum_{i=1}^{5}x_{i}=32 + 20+23 + 26+24=125\). So \(\bar{x}=\frac{125}{5}=25\).

Step2: Calculate the standard - deviation of the original data list

The formula for the standard - deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\).
\((32 - 25)^{2}=49\), \((20 - 25)^{2}=25\), \((23 - 25)^{2}=4\), \((26 - 25)^{2}=1\), \((24 - 25)^{2}=1\).
\(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=49+25 + 4+1+1=80\). Then \(s=\sqrt{\frac{80}{4}}=\sqrt{20}\approx4.47\).

Step3: Convert each value to standard units (z - scores)

The formula for the z - score is \(z=\frac{x-\bar{x}}{s}\).
For \(x = 32\), \(z=\frac{32 - 25}{4.47}\approx1.57\); for \(x = 20\), \(z=\frac{20 - 25}{4.47}\approx - 1.12\); for \(x = 23\), \(z=\frac{23 - 25}{4.47}\approx - 0.45\); for \(x = 26\), \(z=\frac{26 - 25}{4.47}\approx0.22\); for \(x = 24\), \(z=\frac{24 - 25}{4.47}\approx - 0.22\).

Step4: Calculate the average of the converted list

The z - scores are \(1.57,-1.12,-0.45,0.22,-0.22\). \(\sum_{i = 1}^{5}z_{i}=1.57-1.12 - 0.45+0.22-0.22 = 0\). The average of the z - scores \(\bar{z}=\frac{\sum_{i = 1}^{5}z_{i}}{5}=\frac{0}{5}=0\).

Step5: Calculate the standard - deviation of the converted list

The formula for the standard - deviation of the z - scores \(s_{z}=\sqrt{\frac{\sum_{i = 1}^{n}(z_{i}-\bar{z})^{2}}{n - 1}}\). Since \(\bar{z}=0\), \(\sum_{i = 1}^{5}(z_{i}-0)^{2}=(1.57)^{2}+(-1.12)^{2}+(-0.45)^{2}+(0.22)^{2}+(-0.22)^{2}=2.4649 + 1.2544+0.2025 + 0.0484+0.0484 = 4\). Then \(s_{z}=\sqrt{\frac{4}{4}} = 1\).

Answer:

a. 25
b. 4.47
c. \(z_1\approx1.57,z_2\approx - 1.12,z_3\approx - 0.45,z_4\approx0.22,z_5\approx - 0.22\)
d. 0
e. 1