To find the standard deviation of the given dataset, we can follow these steps:
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Calculate the Mean (Average):
- Add up all the numbers.
- Divide by the total number of values in the dataset.
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Calculate the Variance:
- Subtract the mean from each number to find the deviation from the mean.
- Square each deviation.
- Find the average of those squared deviations to get the variance.
-
Calculate the Standard Deviation:
- Take the square root of the variance.
Step 1: Calculate the Mean
Given data:
34, 45, 50, 56, 60, 65, 70, 70, 70, 70, 72, 77, 77, 80, 82, 86, 86, 89, 95, 98
Total numbers: 20
Sum of the numbers:
34 + 45 + 50 + 56 + 60 + 65 + 70 + 70 + 70 + 70 + 72 + 77 + 77 + 80 + 82 + 86 + 86 + 89 + 95 + 98 = 1365
Mean = Sum / Total numbers = 1365 / 20 = 68.25
Step 2: Calculate the Variance
Now we calculate the squared deviations from the mean for each data point:
- (34 - 68.25)² = 1187.5625
- (45 - 68.25)² = 552.5625
- (50 - 68.25)² = 330.0625
- (56 - 68.25)² = 148.5625
- (60 - 68.25)² = 67.5625
- (65 - 68.25)² = 10.5625
- (70 - 68.25)² = 3.0625
- (70 - 68.25)² = 3.0625
- (70 - 68.25)² = 3.0625
- (70 - 68.25)² = 3.0625
- (72 - 68.25)² = 14.0625
- (77 - 68.25)² = 77.5625
- (77 - 68.25)² = 77.5625
- (80 - 68.25)² = 139.0625
- (82 - 68.25)² = 188.0625
- (86 - 68.25)² = 313.5625
- (86 - 68.25)² = 313.5625
- (89 - 68.25)² = 427.5625
- (95 - 68.25)² = 710.0625
- (98 - 68.25)² = 903.5625
Now sum these squared deviations:
Sum of squared deviations =
1187.5625 + 552.5625 + 330.0625 + 148.5625 + 67.5625 + 10.5625 + 3.0625
+ 3.0625 + 3.0625 + 3.0625 + 14.0625 + 77.5625 + 77.5625 + 139.0625
+ 188.0625 + 313.5625 + 313.5625 + 427.5625 + 710.0625 + 903.5625
= 4425.0
Variance (σ²) = Sum of squared deviations / Total numbers Variance = 4425.0 / 20 = 221.25
Step 3: Calculate the Standard Deviation
Standard Deviation (σ) = √Variance σ = √221.25 ≈ 14.87
So, the standard deviation of the given dataset is approximately 14.87.