To calculate the standard deviation of the given scores, we'll follow these steps:
1. Calculate the Mean (Average) of the Scores
2. Calculate the Differences from the Mean for Each Score
3. Square Each Difference
4. Calculate the Mean of These Squared Differences (Variance)
5. Take the Square Root of the Variance (Standard Deviation)
### Step 1: Calculate the Mean
The mean (average) is calculated by summing all the scores and then dividing by the number of scores.
The scores are: 10, 12, 15, 35, 40, 43, 47, 49, 50, 55
Mean = (10 + 12 + 15 + 35 + 40 + 43 + 47 + 49 + 50 + 55) / 10
Mean = 356 / 10
Mean = 35.6
### Step 2: Calculate the Differences from the Mean for Each Score
Now, subtract the mean from each score to find the difference.
- (10 - 35.6) = -25.6
- (12 - 35.6) = -23.6
- (15 - 35.6) = -20.6
- (35 - 35.6) = -0.6
- (40 - 35.6) = 4.4
- (43 - 35.6) = 7.4
- (47 - 35.6) = 11.4
- (49 - 35.6) = 13.4
- (50 - 35.6) = 14.4
- (55 - 35.6) = 19.4
### Step 3: Square Each Difference
Square each of the differences calculated in Step 2.
- (-25.6)^2 = 655.36
- (-23.6)^2 = 556.96
- (-20.6)^2 = 424.36
- (-0.6)^2 = 0.36
- 4.4^2 = 19.36
- 7.4^2 = 54.76
- 11.4^2 = 129.96
- 13.4^2 = 179.56
- 14.4^2 = 207.36
- 19.4^2 = 376.36
### Step 4: Calculate the Mean of These Squared Differences (Variance)
Sum up all the squared differences and then divide by the number of scores to get the variance.
Variance = (655.36 + 556.96 + 424.36 + 0.36 + 19.36 + 54.76 + 129.96 + 179.56 + 207.36 + 376.36) / 10
Variance = 2604.4 / 10
Variance = 260.44
### Step 5: Take the Square Root of the Variance (Standard Deviation)
Finally, take the square root of the variance to get the standard deviation.
Standard Deviation = √260.44
Standard Deviation ≈ 16.14
So, the mean of the scores is 35.6, the variance is 260.44, and the standard deviation is approximately 16.14.
