Answer :
To find the sample standard deviation for the given data set, follow these steps:
1. List the Data: Start with the data set provided: 86, 83, 76, 82, 67, 69, 82, 82, 69.
2. Calculate the Mean:
- First, sum all the numbers: 86 + 83 + 76 + 82 + 67 + 69 + 82 + 82 + 69 = 696.
- Next, divide the sum by the number of values to find the mean: 696 / 9 = 77.33 (to two decimal places).
3. Find Each Deviation from the Mean:
- Subtract the mean from each data value to find the deviations:
- 86 - 77.33 = 8.67
- 83 - 77.33 = 5.67
- 76 - 77.33 = -1.33
- 82 - 77.33 = 4.67
- 67 - 77.33 = -10.33
- 69 - 77.33 = -8.33
- 82 - 77.33 = 4.67
- 82 - 77.33 = 4.67
- 69 - 77.33 = -8.33
4. Square Each Deviation:
- Convert each deviation to a squared value:
- 8.67² = 75.23
- 5.67² = 32.14
- (-1.33)² = 1.77
- 4.67² = 21.81
- (-10.33)² = 106.69
- (-8.33)² = 69.39
- 4.67² = 21.81
- 4.67² = 21.81
- (-8.33)² = 69.39
5. Calculate the Variance:
- Find the average of these squared differences, adjusting for sample variance:
- Sum of squared differences: 75.23 + 32.14 + 1.77 + 21.81 + 106.69 + 69.39 + 21.81 + 21.81 + 69.39 = 420.
- Divide by the number of data points minus one (N - 1): 420 / (9 - 1) = 52.5.
6. Calculate the Standard Deviation:
- The sample standard deviation is the square root of the variance:
- √52.5 = 7.246 (rounded to the nearest thousandth).
So, the sample standard deviation for the given data set is approximately 7.246.
1. List the Data: Start with the data set provided: 86, 83, 76, 82, 67, 69, 82, 82, 69.
2. Calculate the Mean:
- First, sum all the numbers: 86 + 83 + 76 + 82 + 67 + 69 + 82 + 82 + 69 = 696.
- Next, divide the sum by the number of values to find the mean: 696 / 9 = 77.33 (to two decimal places).
3. Find Each Deviation from the Mean:
- Subtract the mean from each data value to find the deviations:
- 86 - 77.33 = 8.67
- 83 - 77.33 = 5.67
- 76 - 77.33 = -1.33
- 82 - 77.33 = 4.67
- 67 - 77.33 = -10.33
- 69 - 77.33 = -8.33
- 82 - 77.33 = 4.67
- 82 - 77.33 = 4.67
- 69 - 77.33 = -8.33
4. Square Each Deviation:
- Convert each deviation to a squared value:
- 8.67² = 75.23
- 5.67² = 32.14
- (-1.33)² = 1.77
- 4.67² = 21.81
- (-10.33)² = 106.69
- (-8.33)² = 69.39
- 4.67² = 21.81
- 4.67² = 21.81
- (-8.33)² = 69.39
5. Calculate the Variance:
- Find the average of these squared differences, adjusting for sample variance:
- Sum of squared differences: 75.23 + 32.14 + 1.77 + 21.81 + 106.69 + 69.39 + 21.81 + 21.81 + 69.39 = 420.
- Divide by the number of data points minus one (N - 1): 420 / (9 - 1) = 52.5.
6. Calculate the Standard Deviation:
- The sample standard deviation is the square root of the variance:
- √52.5 = 7.246 (rounded to the nearest thousandth).
So, the sample standard deviation for the given data set is approximately 7.246.
