Description

Chamberlain University

MATH 225N- Statistical Reasoning for the health Sciences

Week 3: Standard Deviation

Question

A company is interested to know the variation in yearly sales amount for all 5 salespeople in the company.

The dataset shown below is the sales amount sold by the 5 salespeople in the company (expressed in thousands of dollars):

40,60,65,70,80

Find the variance for this dataset.

Answer

Notice the dataset represents the entire population of sales amount for the company.  This indicates that the variance will reflect the population variance.

To calculate the population variance, first find the mean

40+60+65+70+805=315=63

5                        5

Next, subtract the mean from each salary and then square each result.  Then add up these squared quantities and divide by the number of salespeople.

(40−63)²+(60−63)²+(65−63)²+(70−63)²+(80−63)²

5

(−23)²+(−3)²+2²+7²+17²

5

= 880/5

=176

The variance is 176.

 

Question

Find the sample variance of the following set of data:
14, 10, 9, 7, 10.

• Round the final answer to one decimal place.

First, we find that the mean is

14+10+9+7+10

5

=50/5

=10

Now, we need to take the deviations from the mean and square them:

 

Value	Deviation	Deviation2
14	4.0	16.0
10	0.0	0.0
9	-1.0	1.0
7	-3.0	9.0
10	0.0	0.0

 

Finally, we add up the squared deviations and divide by the number of data values minus one (5−1=4).

(16.0+0.0+1.0+9.0+0.0)/4

=6.5

Question

Using the following set of data (the same as in the previous problem), find the sample standard deviation:
14, 10, 9, 7, 10.

• Round the final answer to one decimal place.

Remember that the sample standard deviation is the square root of the sample variance. Since we just found that the sample variance is 6.5, we find that the sample standard deviation is √6.5 =2.5.

 

Understand the standard deviation of a set of data

Question

Which of the following lists of data has the smallest standard deviation?

• 28, 26, 20, 17, 21, 29, 28, 28, 17, 22

 

• 14, 15, 15, 12, 11, 14, 11, 13, 14, 12

 

• 34, 26, 34, 26, 22, 34, 24, 26, 25, 24

 

• 21, 15, 14, 27, 21, 24, 27, 20, 20, 30

 

• 9, 17, 21, 9, 14, 18, 22, 10, 12, 16

Answer Explanation

Correct answer:

14, 15, 15, 12, 11, 14, 11, 13, 14, 12

Remember that standard deviation is a measure of how spread out the values are. The list 14, 15, 15, 12, 11, 14, 11, 13, 14, 12 has the smallest standard deviation because its values are all relatively close together.

Question

Which of the data sets represented by the following box and whisker plots has the largest standard deviation?

Four horizontal box-and-whisker plots share a vertical axis with the classes D, C, B, and A and a horizontal axis from 0 to 120 in increments of 20. The box-and-whisker plot above the class label A has the following five-number summary: 27, 59, 70, 81, and 120. The box-and-whisker plot above the class label B has the following five-number summary: 55, 65, 69, 73, and 88. The box-and-whisker plot above the class label C has the following five-number summary: 35, 62, 69, 78, and 102. The box-and-whisker plot above the class label D has the following five-number summary: 10, 59, 70, 81, and 138. All values are approximate.

Answer Explanation

Correct answer:

D

Remember that the standard deviation is a measure of how spread out the data is. If the values are concentrated around the mean, then a data set has a lower standard deviation.

Therefore, a box and whisker plot with long whiskers and a long box has values that are more spread out, and hence has a larger standard deviation.

Question

The following data values represent the daily amount spent by a family during a 7 day summer vacation.

Find the population standard deviation of this dataset:

$96,$125,$80,$110,$75,$100,$121

(Round your answer to 1 decimal place).

Answer Explanation

Correct answers:

• σ=17.7

Notice the dataset represents the entire population of expenses for the 7 day vacation.
To determine the standard deviation for population data:

1. Find the mean of the dataset

(96+125+80+110+75+100+121)/ 7

=101

1. Subtract the mean from each data value and square each result.

x	x−x¯	(x−x¯)2
96	−5	25
125	24	576
80	−21	441
110	9	81
75	−26	676
100	−1	1
121	20	400

 

1. Add up the squared quantities from Step 2.

25+576+441+81+676+1+400=2200

1. Divide this sum from Step 3 by the number of data values. This is the variance σ2.

σ2=2200 / 7

1. Take the square root of the result from Step 4.  This result represents the population standard deviation.

σ =√2200/7

≈17.7

 

Question

Imagine customers waiting in line for tellers at two different banks. Customers at Bank of Jehovah can enter any one of three different lines leading to three different tellers. ARAC Bank also has three different tellers, but all customers wait in a single line and are called to the next available teller.  The following data sets include the waiting times (in minutes) for eleven customers at each bank. Which bank do you think has the least amount of variation in waiting times?

Bank of Jehovah	5.1,6.2,6.6,7.2,7.7,8.2,8.7,8.7,9.5,10.3,11.0
ARAC Bank	7.6,7.7,7.7,7.9,8.1,8.2,8.3,8.4,8.7,8.8,8.8

 

• Bank of Jehovah

 

• ARAC Bank

Answer Explanation

Correct answer:

ARAC Bank

Remember that standard deviation is a measure of how spread out the values are. We can get a very rough estimate of the standard deviation for each data set using the formula for the range rule of thumb:

Standard deviation ≈ range ≈ maximum value−minimum value

4                                   4

The data set with the smallest result will have the smallest standard deviation and the least spread. The data set with the largest result will have the largest standard deviation and the most spread.

Bank of Jehovah: Estimation of standard deviation

≈ (11.0−5.1)/4

≈ 5.9 /4

≈1.475

ARAC Bank: Estimation of standard deviation

≈ (8.8−7.6)/4

≈1.2/4

≈0.3

We see that ARAC Bank consisting of 7.6,7.7,7.7,7.9,8.1,8.2,8.3,8.4,8.7,8.8,8.8_ has the smallest estimated standard deviation. Therefore, ARAC Bank has the least amount of variation or spread in its wait times.

Question

Body mass index (BMI) is a medical screening tool that measures body fat in an individual based on height and weight. The following data sets include the BMI measurements for random samples of 7 individuals. Use the range rule of thumb to estimate the standard deviation, then determine which sample has the least amount of variation.

Sample 1	25.8,39.2,32.2,18.7,22.7,28.8,25.4
Sample 2	24.5,33.3,31.7,45.9,23.4,28.2,25.8
Sample 3	30.8,19.2,29.9,25.6,28.8,31.0,27.3
Sample 4	31.8,23.9,29.1,52.6,23.6,31.1,20.6

 

• Sample 1
• Sample 2
• Sample 3
• Sample 4

Answer Explanation

Correct answer:

Sample 3

Remember that standard deviation is a measure of how spread out the values are. We can get a very rough estimate of the standard deviation for each data set using the formula for the range rule of thumb:

Standard deviation ≈ range ≈ maximum value – minimum value

4                                          4

The data set with the smallest result will have the smallest standard deviation and the least spread. The data set with the largest result will have the largest standard deviation and the most spread.

Sample 1: Estimation of standard deviation

≈ (39.2−18.7) / 4

≈20.5/ 4

≈5.125

Sample 2: Estimation of standard deviation

≈ (45.9−23.4)/ 4

≈ 22.5/ 4

≈ 5.625

Sample 3: Estimation of standard deviation

≈ (31.0−19.2)/ 4

≈11.8/ 4

≈2.95

Sample 4: Estimation of standard deviation

≈ (52.6−20.6)/4

≈32.0/4

≈8.0

We see that Sample 3 consisting of 30.8,19.2,29.9,25.6,28.8,31.0,27.3 has the smallest estimated standard deviation. Therefore, Sample 3 has the least amount of variation or spread.

 

For any assistance in this course or any other, please contact us on WhatsApp +254716353533