Question

hw 02 ch 5: problem 3 (12 points)
for the following list of data values: 32, 20, 23, 26, 24
a. find the average. answer:
b. find the sd. answer:
c. convert each value to standard units using the average and sd of the list:
i. 32 z =
ii. 20 z =
iii. 23 z =
iv. 26 z =
v. 24 z =
d. find the average of the converted list. avg_z =
e. find the sd of the converted list. sd_z =
note: you can earn partial credit on this problem.
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Explanation:

Step1: Calculate the average

The formula for the average (mean) $\bar{x}$ of a data - set $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here, $n = 5$, $x_1=32,x_2 = 20,x_3=23,x_4=26,x_5=24$. So, $\sum_{i=1}^{5}x_i=32 + 20+23+26+24=125$, and $\bar{x}=\frac{125}{5}=25$.

Step2: Calculate the standard deviation (SD)

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

Step3: Convert to standard units

The formula for converting a value $x$ to standard units (z - score) is $z=\frac{x-\bar{x}}{s}$.
i. For $x = 32$, $z=\frac{32 - 25}{4.47}\approx1.57$.
ii. For $x = 20$, $z=\frac{20 - 25}{4.47}\approx - 1.12$.
iii. For $x = 23$, $z=\frac{23 - 25}{4.47}\approx - 0.45$.
iv. For $x = 26$, $z=\frac{26 - 25}{4.47}\approx0.22$.
v. For $x = 24$, $z=\frac{24 - 25}{4.47}\approx - 0.22$.

Step4: Calculate the average of the converted list

The average of a set of z - scores is always 0. Mathematically, if $z_i=\frac{x_i-\bar{x}}{s}$ for $i = 1,\cdots,n$, then $\bar{z}=\frac{\sum_{i = 1}^{n}z_i}{n}=\frac{\sum_{i = 1}^{n}\frac{x_i-\bar{x}}{s}}{n}=\frac{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})}{s}}{n}$. Since $\sum_{i = 1}^{n}(x_i-\bar{x})=0$, $\bar{z}=0$.

Step5: Calculate the SD of the converted list

The standard deviation of a set of z - scores is always 1.

Answer:

a. 25
b. $\sqrt{20}\approx4.47$
c. i. 1.57
ii. - 1.12
iii. - 0.45
iv. 0.22
v. - 0.22
d. 0
e. 1