Chapter 3.2, Problem 18E

Use the data from Exercises 7, 15, and 17 (unemployment, prisoners, precipitation days) and compare the standard deviation with that obtained by the range rule of thumb (R/4.) Comment on the results.

Traveler Spending The data show the traveler spending in billions of dollars for a recent year for a sample of the states. Find the range, variance, and standard deviation for the data.

20.1	33.5	21.7	58.4	23.2	110.8	30.9	24.0
74.8	60.0

Source: U.S. Travel Agency.

Laws Passed The data show the number of public laws passed by the U.S. Congress for a sample of recent years. Find the range, variance, and standard deviation for the data.

283 394 383 580 498 460 377 482

Source: Congressional Record.

Annual Precipitation Days The number of annual precipitation days for one-half of the 50 largest U.S. cities is listed below. Find the range, variance, and standard deviation of the data.

135	128	136	78	116	77	111	79	44	97
116	123	88	102	26	82	156	133	107	35
112	98	45	122	125

To determine

The standard deviation by standard formula and by range rule of thumb for the given data and compare the results.

Answer to Problem 18E

The standard deviation for first second and third data is $29.86,91.5$ and $34.01$ respectively. The standard deviation for first second and third data by range rule of thumb is $22.675,74.25$ and $32.5$ respectively. The standard deviation by range rule of thumb is an approximate result, it is not appropriate.

Explanation of Solution

Given info:

Refer to Exercise 7,15 and 17.

The data shows the unemployment as given below:

20.1	33.5	21.7	58.4
23.2	110.8	30.9	24.0
74.8	60.0

The data shows the prisoners as given below:

283

394

383

580

498

460

377

482

The data shows the precipitation days as given below:

135	128	136	78	116	77	111	79	44	97
116	123	88	102	26	82	156	133	107	35
112	98	45	122	125

Calculation:

The formula for standard deviation is,

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

Construct the table to find the above values.

S. no.	Values	$X-\overline{X}$	${\left(X-\overline{X}\right)}^2$
1	20.1	$-25.64$	657.4096
2	33.5	$-12.24$	149.8176
3	21.7	$-24.04$	577.9216
4	58.4	12.66	160.2756
5	23.2	$-22.54$	508.0516
6	110.8	65.06	4232.804
7	30.9	$-14.84$	220.2256
8	24.0	$-21.74$	472.6276
9	74.8	29.06	844.4836
10	60.0	14.26	203.3476
Total	457.4		8026.964

The formula to calculate the mean is,

$\begin{array}{c}\overline{\text{X}}=\frac{{\displaystyle \sum _1^n{X}_n}}{n}\\ =\frac{X_1+X_2+X_3+\mathrm{...}+X_n}{n}\end{array}$ (2)

Where,

• $X$ is the number of observers and active watchers.
• $n$ is the total number of states.

Substitute ${\displaystyle \sum _1^{10}{X}_n}=457.4$ and $n=10$ in equation (2),

$\begin{array}{c}\text{Mean}=\frac{457.4}{10}\\ =45.74\end{array}$

Substitute ${\displaystyle \sum {\left(X-\overline{X}\right)}^2}=8026.964$ and $n=10$ in equation (1),

$\begin{array}{c}s=\sqrt{\frac{8026.964}{10-1}}\\ =\sqrt{\frac{8026.964}{9}}\\ =\sqrt{891.8849}\\ \approx 29.86\end{array}$

The standard deviation for the first data is $29.86$.

Construct the table to find the standard deviation for second data.

S. no.	Values	$X-\overline{X}$	${\left(X-\overline{X}\right)}^2$
1	283	$-149.125$	22238.27
2	394	$-38.125$	1453.516
3	383	$-49.125$	2413.266
4	580	147.875	21867.02
5	498	65.875	4339.516
6	460	27.875	777.0156
7	377	$-55.125$	3038.766
8	482	49.875	2487.516
Total	3457		58614.88

Substitute ${\displaystyle \sum _1^8{X}_n}=34.57$ and $n=8$ in equation (2),

$\begin{array}{c}\text{Mean}=\frac{3457}{8}\\ =432.125\end{array}$

Substitute ${\displaystyle \sum {\left(X-\overline{X}\right)}^2}=58614.88$ and $n=8$ in equation (1),

$\begin{array}{c}s=\sqrt{\frac{58614.88}{8-1}}\\ =\sqrt{\frac{58614.88}{7}}\\ =\sqrt{8373.554}\\ \approx 91.5\end{array}$

The standard deviation for the second data is 91.5.

Construct the table to find the standard deviation for third data.

S. no.	Values	$X-\overline{X}$	${\left(X-\overline{X}\right)}^2$
1	135	36.16	1307.55
2	128	29.16	850.31
3	136	37.16	1380.87
4	78	$-20.84$	434.31
5	116	17.16	294.47
6	77	$-21.84$	476.99
7	111	12.16	147.87
8	79	$-19.84$	393.63
9	44	$-54.84$	3007.43
10	97	$-1.84$	3.39
11	116	17.16	294.47
12	123	24.16	583.71
13	88	$-10.84$	117.51
14	102	3.16	9.99
15	26	$-72.84$	5305.67
16	82	$-16.84$	283.59
17	156	57.16	3267.27
18	133	34.16	1166.91
19	107	8.16	66.59
20	35	$-63.84$	4075.55
21	112	13.16	173.19
22	98	$-0.84$	0.71
23	45	$-53.84$	2898.75
24	122	23.16	536.39
25	125	26.16	684.35
Total	2471		27761.36

Substitute ${\displaystyle \sum _1^{25}{X}_n}=2471$ and $n=25$ in equation (2),

$\begin{array}{c}\text{Mean}=\frac{2471}{25}\\ =98.84\end{array}$

Substitute ${\displaystyle \sum {\left(X-\overline{X}\right)}^2}=27761.36$ and $n=25$ in equation (1),

$\begin{array}{c}s=\sqrt{\frac{27761.36}{25-1}}\\ =\sqrt{\frac{27761.36}{24}}\\ =\sqrt{1156.723}\\ \approx 34.01\end{array}$

The standard deviation for the third data is 34.01.

The formula to calculate the standard deviation by range rule is,

$s\approx \frac{\text{Range}}{4}$ (3)

Where,

• The range is the difference between the largest data value and the smallest data value.

For the unemployment data, the smallest data value is 20.1 and the largest data value is 110.8, therefore the difference between them is 90.7.

Substitute 90.7 for range in equation (3),

$\begin{array}{c}s\approx \frac{90.7}{4}\\ =22.675\end{array}$

The standard deviation for the first data is 22.675.

For the prisoner data, the smallest data value is 283 and the largest data value is 580, therefore the difference between them is 297.

Substitute 297 for range in equation (3),

$\begin{array}{c}s\approx \frac{297}{4}\\ =74.25\end{array}$

The standard deviation for the second data is 74.25.

For the precipitation day data, the smallest data value is 26 and the largest data value is 156, therefore the difference between them is 130.

Substitute 130 for range in equation (3),

$\begin{array}{c}s\approx \frac{130}{4}\\ =32.5\end{array}$

The standard deviation for the third data is 32.5.

Therefore, standard deviation for first second and third data is $29.86,91.5$ and $34.01$ respectively. The standard deviation for first second and third data by range rule of thumb is $22.675,74.25$ and $32.5$ respectively. The standard deviation by range rule of thumb is an approximate result, it is not appropriate.