Sample salaries (in dollars) of high school teachers are listed. Find the sample mean and standard deviation of the data.
49,632
54,619
58,298
48,250
51,842
50,875
43,219
49,924
The sample mean is the sum divided by 8 (the number of people) Call than number Xm.
The standard deviation is the square root of the average of [Salary ^2] - Xm^2.
You could also calculate it as the square root of the average of [Salary -Xm)^2
The first way is faster.
To find the sample mean and standard deviation of the data, follow these steps:
Step 1: Add up all the salaries
49,632 + 54,619 + 58,298 + 48,250 + 51,842 + 50,875 + 43,219 + 49,924 = 406,459
Step 2: Determine the number of salaries in the data
In this case, there are 8 salaries.
Step 3: Calculate the sample mean (average)
Mean = Sum of all salaries / Number of salaries
Mean = 406,459 / 8
Mean ≈ 50,807.375
The sample mean of the given data is approximately $50,807.375.
Step 4: Calculate the sum of squared differences from the mean
Subtract the mean from each salary, square the result, and sum up the squared differences.
Squared difference for each salary:
(49,632 - 50,807.375)² = 140265.5156
(54,619 - 50,807.375)² = 143918.1406
(58,298 - 50,807.375)² = 56490.7656
(48,250 - 50,807.375)² = 65470.0156
(51,842 - 50,807.375)² = 107195.6406
(50,875 - 50,807.375)² = 4504.2656
(43,219 - 50,807.375)² = 56989.0156
(49,924 - 50,807.375)² = 78324.0156
Step 5: Calculate the variance
Variance = Sum of squared differences / (Number of salaries - 1)
Variance = (140265.5156 + 143918.1406 + 56490.7656 + 65470.0156 + 107195.6406 + 4504.2656 + 56989.0156 + 78324.0156) / (8-1)
Variance ≈ 413620.1395
Step 6: Calculate the standard deviation
Standard Deviation = square root of the variance
Standard Deviation ≈ sqrt(413620.1395)
Standard Deviation ≈ 642.705
Therefore, the sample standard deviation of the given data is approximately $642.705.