The data shows the total number of employee medical leave days taken for on-the-job accidents in the first six months of the year: 16, 7, 23, 9, 25, 16. Find the standard deviation.
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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.
To find the standard deviation, you will need to follow these steps:
Step 1: Find the mean (average) of the data set.
Step 2: For each data point, subtract the mean and square the result.
Step 3: Find the mean (average) of the squared differences.
Step 4: Take the square root of the mean squared differences from step 3.
Let's go through these steps one by one.
Step 1: Find the mean
To find the mean, add up all the data points and divide the sum by the number of data points:
Mean = (16 + 7 + 23 + 9 + 25 + 16) / 6 = 96 / 6 = 16
Step 2: Calculate the squared differences
For each data point, subtract the mean and square the result:
(16 - 16)^2 = 0^2 = 0
(7 - 16)^2 = (-9)^2 = 81
(23 - 16)^2 = 7^2 = 49
(9 - 16)^2 = (-7)^2 = 49
(25 - 16)^2 = 9^2 = 81
(16 - 16)^2 = 0^2 = 0
Step 3: Find the mean of the squared differences
Add up all the squared differences and divide by the number of data points:
Mean squared differences = (0 + 81 + 49 + 49 + 81 + 0) / 6 = 260 / 6 = 43.33
Step 4: Take the square root
Take the square root of the mean squared differences:
Standard deviation = √43.33 ≈ 6.58
Therefore, the standard deviation for the given data set is approximately 6.58.
To find the standard deviation of a set of data, you can follow these steps:
1. Calculate the mean (average) of the data set.
To do this, add up all the values in the data set and divide by the number of values. In this case, the sum is 16 + 7 + 23 + 9 + 25 + 16 = 96, and there are 6 values. So the mean is 96 / 6 = 16.
2. Subtract the mean from each value in the data set.
For each value in the data set, subtract the mean calculated in step 1. The resulting values are called the deviations from the mean.
Deviations: (16 - 16), (7 - 16), (23 - 16), (9 - 16), (25 - 16), (16 - 16)
Deviations: 0, -9, 7, -7, 9, 0
3. Square each deviation.
Square each deviation calculated in step 2. This step is necessary because deviations can be positive or negative, and squaring them ensures that all values are positive.
Squared deviations: 0^2, (-9)^2, 7^2, (-7)^2, 9^2, 0^2
Squared deviations: 0, 81, 49, 49, 81, 0
4. Calculate the mean of the squared deviations.
Find the mean of the squared deviations by adding them up and dividing by the number of values. In this case, the sum is 0 + 81 + 49 + 49 + 81 + 0 = 260, and there are 6 values. So the mean of the squared deviations is 260 / 6 = 43.33 (rounded to two decimal places).
5. Take the square root of the mean of the squared deviations.
Finally, take the square root of the mean of the squared deviations calculated in step 4. This will give you the standard deviation, which measures the dispersion or spread of the data set.
Standard deviation: √43.33 ≈ 6.58 (rounded to two decimal places).
Therefore, the standard deviation of the employee medical leave days taken for on-the-job accidents in the first six months of the year is approximately 6.58.