Question

34,45,50,56,60,65,70,70,70,70,72,77,77,80,82,86,86,89,95,98
find the one standard deviation

Answer 1

To find the standard deviation of the given dataset, we can follow these steps:

1. **Calculate the Mean** (Average):
- Add up all the numbers.
- Divide by the total number of values in the dataset.

2. **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.

3. **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**.