Chapter 8, Problem 7TP

To determine

The standard score for each of the score by SPSS and manually given in the data set.

Answer to Problem 7TP

The Z score for each score obtained by SPSS is shown below.

Fig (1)

The Z score for each scores obtained manual is shown in table below.

$\text{X}$	$\left(\text{X}-\overline{\text{X}}\right)$	${\left(\text{X}-\overline{\text{X}}\right)}^2$	Z score
18	$-7.7333$	59.8044	$\frac{-7.733}{7.63}=-1.012$
19	$-6.7333$	45.3378	$\frac{-6.7333}{7.63}=-0.881$
15	$\begin{array}{c}-10.733\end{array}$	115.204	$\frac{-10.733}{7.63}=--1.405$
20	$\begin{array}{c}-5.7333\end{array}$	32.8711	$\frac{-5.733}{7.63}=-0.751$
25	$-0.7333$	0.53778	$\frac{-0.7333}{7.63}=--0.096$
31	5.26667	27.7378	$\frac{5.266}{7.62}=0.689$
17	$\begin{array}{c}-8.7333\end{array}$	76.2711	$\frac{-8.733}{7.63}=-1.143$
35	9.26667	85.8711	$\frac{9.266}{7.63}=1.213$
27	1.26667	1.60444	$\frac{1.266}{7.63}=0.166$
22	$-3.7333$	13.9378	$\frac{-3.733}{7.63}=-0.489$
34	8.26667	68.3378	$\frac{8.266}{7.63}=1.082$
29	3.26667	10.6711	$\frac{3.266}{7.63}=0.428$
40	14.2667	203.538	$\frac{14.266}{7.63}=1.868$
33	7.26667	52.8044	$\frac{7.266}{7.63}=0.951$
21	$\begin{array}{c}-4.7333\end{array}$	22.4044	$\frac{-4.733}{7.63}=-0.62$
Total		816.933

Fig (2)

The above both tables show same Z scores for each of the score.

Explanation of Solution

Given info:

The data set is given as follows:

Score
18
19
15
20
25
31
17
35
27
22
34
29
40
33
21

Calculation:

A z-score is the quantity of standard deviations from the mean an information point is. In any case, more in fact it's a measure of what numbers of standard deviations underneath or over the populace mean a crude score is. A z-score is otherwise called a standard score.

Step by step procedure to obtain the Z score using SPSS software is given below:

Steps for calculating Z score in SPSS:

• Go to Variable View, in first row write Score and come back to Data View.
• Click Analyze $\to$ Descriptive Statistics $\to$ Descriptive.
• Double click on the variable score to move it to the variables $\to$ Tick on Save Standardized values as variables.
• Click ok.
• The output will be as shown below.

Output using the SPSS software is given below:

Fig (1)

The above table shows the Z score corresponding to the each score.

The formula to calculate Z score is,

$Z=\frac{\text{X}-\overline{\text{X}}}{s}$

Mean:

The formula to calculate mean is,

$\begin{array}{c}\overline{\text{X}}=\frac{{\displaystyle \sum \text{X}}}{n}\\ {\displaystyle \sum \text{X}}=\left\{\begin{array}{l}18+19+15+20+25+31+17+35+\\ 27+22+34+29+40+33+21\end{array}\right\}\\ =386\end{array}$

Substitute 386 for ${\displaystyle \sum \text{X}}$ and 15 for n in the above equation (1).

$\begin{array}{c}\overline{\text{X}}=\frac{386}{15}\\ =25.733\end{array}$

The formula to calculate sample standard deviation is,

$s=\sqrt{\frac{1}{n-1}{{\displaystyle \sum \left(\text{X}-\overline{\text{X}}\right)}}^2}$ (2)

Substitute 816.933 for ${\displaystyle \sum {\left(\text{X}-\overline{\text{X}}\right)}^2}$ and 15 for n in the above equation (2).

$\begin{array}{c}s=\sqrt{\frac{1}{15-1}\times \left(816.933\right)}\\ =\sqrt{58.35}\\ =7.63\end{array}$

Use sample standard deviation for calculating of Z score and make a table of values.

$\text{X}$	$\left(\text{X}-\overline{\text{X}}\right)$	${\left(\text{X}-\overline{\text{X}}\right)}^2$	Z score
18	$-7.7333$	59.8044	$\frac{-7.733}{7.63}=-1.012$
19	$-6.7333$	45.3378	$\frac{-6.7333}{7.63}=-0.881$
15	$\begin{array}{c}-10.733\end{array}$	115.204	$\frac{-10.733}{7.63}=--1.405$
20	$\begin{array}{c}-5.7333\end{array}$	32.8711	$\frac{-5.733}{7.63}=-0.751$
25	$-0.7333$	0.53778	$\frac{-0.7333}{7.63}=--0.096$
31	5.26667	27.7378	$\frac{5.266}{7.62}=0.689$
17	$\begin{array}{c}-8.7333\end{array}$	76.2711	$\frac{-8.733}{7.63}=-1.143$
35	9.26667	85.8711	$\frac{9.266}{7.63}=1.213$
27	1.26667	1.60444	$\frac{1.266}{7.63}=0.166$
22	$-3.7333$	13.9378	$\frac{-3.733}{7.63}=-0.489$
34	8.26667	68.3378	$\frac{8.266}{7.63}=1.082$
29	3.26667	10.6711	$\frac{3.266}{7.63}=0.428$
40	14.2667	203.538	$\frac{14.266}{7.63}=1.868$
33	7.26667	52.8044	$\frac{7.266}{7.63}=0.951$
21	$\begin{array}{c}-4.7333\end{array}$	22.4044	$\frac{-4.733}{7.63}=-0.62$
Total		816.933

Fig (1)

There is no difference of Z score whether Z score calculated by SPSS and manually.