98,99,99,103,101,102,104,89,87,90,90
Within how many standard deviations of the mean is the value 102?
To determine the number of standard deviations from the mean, we need to calculate the mean and standard deviation. Here's how to do it step by step:
Step 1: Calculate the mean:
Add up all the values in the dataset and divide it by the total number of values.
Mean = (98 + 99 + 99 + 103 + 101 + 102 + 104 + 89 + 87 + 90 + 90) / 11 = 1,052 / 11 = 95.636
Step 2: Calculate the standard deviation:
The formula for calculating the standard deviation for a sample is as follows:
s = sqrt((Σ(xᵢ - x̄)²) / (n - 1))
Where,
s = standard deviation
Σ = sum of (xᵢ - x̄)²
xᵢ = individual values
x̄ = mean of the dataset
n = total number of values
Let's calculate the standard deviation step by step:
1. Find the deviations from the mean for each value by subtracting the mean from each value:
(98 - 95.636)², (99 - 95.636)², (99 - 95.636)², (103 - 95.636)², (101 - 95.636)², (102 - 95.636)², (104 - 95.636)², (89 - 95.636)², (87 - 95.636)², (90 - 95.636)², (90 - 95.636)²
2. Square each deviation:
(2.364², 3.364², 3.364², 7.364², 5.364², 6.364², 8.364², 6.636², 8.636², 5.636², 5.636²)
3. Sum up all the squared deviations:
2.364² + 3.364² + 3.364² + 7.364² + 5.364² + 6.364² + 8.364² + 6.636² + 8.636² + 5.636² + 5.636² = 269.091
4. Divide the sum of squared deviations by (n - 1):
269.091 / (11 - 1) = 269.091 / 10 = 26.9091
5. Take the square root of the result to get the standard deviation:
s = sqrt(26.9091) ≈ 5.1884
Step 3: Calculate the number of standard deviations from the mean:
To calculate the number of standard deviations from the mean for a given value, we use the formula:
Number of standard deviations = (Value - Mean) / Standard Deviation
For the value 102:
Number of standard deviations = (102 - 95.636) / 5.1884 ≈ 1.221
Therefore, the value 102 is approximately 1.221 standard deviations away from the mean.