The monthly salaries (in thousands of dollars) of a sample of sales representatives are as follows:
13, 7, 10, 9, 8, 13
Find the standard deviation of this sample of monthly salaries. Round your answer to at least two decimal places.
To find the standard deviation of a sample, follow these steps:
1. Find the mean (average) of the sample.
2. Subtract the mean from each value in the sample.
3. Square each of the differences.
4. Find the mean of the squared differences.
5. Take the square root of the mean of the squared differences.
Let's calculate the standard deviation step by step:
Step 1: Find the Mean
Add up all the salaries: 13 + 7 + 10 + 9 + 8 + 13 = 60
Divide the sum by the number of salaries: 60 / 6 = 10
The mean of the sample is 10.
Step 2: Subtract the Mean from Each Value
Subtract the mean (10) from each value in the sample:
13 - 10 = 3
7 - 10 = -3
10 - 10 = 0
9 - 10 = -1
8 - 10 = -2
13 - 10 = 3
The differences are: 3, -3, 0, -1, -2, 3.
Step 3: Square Each of the Differences
Square each of the differences:
3^2 = 9
(-3)^2 = 9
0^2 = 0
(-1)^2 = 1
(-2)^2 = 4
3^2 = 9
The squared differences are: 9, 9, 0, 1, 4, 9.
Step 4: Find the Mean of the Squared Differences
Add up all the squared differences: 9 + 9 + 0 + 1 + 4 + 9 = 32
Divide the sum by the number of squared differences (6): 32 / 6 = 5.33
The mean of the squared differences is approximately 5.33.
Step 5: Take the Square Root of the Mean of the Squared Differences
Take the square root of the mean of the squared differences: √5.33 ≈ 2.31
The standard deviation of the sample of monthly salaries is approximately 2.31 (thousands of dollars), rounded to two decimal places.
To find the standard deviation of this sample of monthly salaries, you can follow these steps:
1. Calculate the mean (average) of the sample. To do this, add up all the salaries and divide the sum by the number of salaries in the sample.
13 + 7 + 10 + 9 + 8 + 13 = 60
60 / 6 = 10
2. Calculate the deviation of each salary from the mean. To do this, subtract the mean from each salary.
Deviation = Salary - Mean
Deviation for 13 = 13 - 10 = 3
Deviation for 7 = 7 - 10 = -3
Deviation for 10 = 10 - 10 = 0
Deviation for 9 = 9 - 10 = -1
Deviation for 8 = 8 - 10 = -2
Deviation for 13 = 13 - 10 = 3
3. Square each deviation. This step is necessary because deviations can be positive or negative, and squaring them makes them all positive.
Squared Deviation for 13 = 3^2 = 9
Squared Deviation for 7 = (-3)^2 = 9
Squared Deviation for 10 = 0^2 = 0
Squared Deviation for 9 = (-1)^2 = 1
Squared Deviation for 8 = (-2)^2 = 4
Squared Deviation for 13 = 3^2 = 9
4. Calculate the average of the squared deviations. This is the variance of the sample.
Variance = (Sum of Squared Deviations) / (Number of Salaries)
(9 + 9 + 0 + 1 + 4 + 9) / 6 = 32 / 6 = 5.33
5. Find the square root of the variance to get the standard deviation.
Standard Deviation = √Variance
Standard Deviation = √5.33 ≈ 2.31 (rounded to two decimal places)
Therefore, the standard deviation of this sample of monthly salaries is approximately 2.31 thousand dollars.
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.