a sample consists of the following n =6 scores: 2, 7, 4, 6, 4, and 7 compute the mean and standard deviation for the sample
To compute the mean and standard deviation for the given sample, you need to follow a series of steps.
Step 1: Calculate the Mean
The mean, or average, is calculated by summing up all the scores in the sample and then dividing by the number of scores.
For the given sample: 2, 7, 4, 6, 4, and 7
Add up all the scores: 2 + 7 + 4 + 6 + 4 + 7 = 30
Divide the sum by the number of scores (n = 6): 30 ÷ 6 = 5
Hence, the mean of the sample is 5.
Step 2: Calculate the Deviations from the Mean
To find the standard deviation, we first need to find the individual deviations from the mean for each score in the sample. The deviation from the mean is calculated by subtracting the mean from each score.
For the given sample, subtract the mean (5) from each score:
2 - 5 = -3
7 - 5 = 2
4 - 5 = -1
6 - 5 = 1
4 - 5 = -1
7 - 5 = 2
The deviations from the mean are as follows: -3, 2, -1, 1, -1, and 2.
Step 3: Calculate the Variance
The standard deviation is computed using the variance. To find the variance, follow these steps:
1. Square each deviation from the mean.
2. Sum up all the squared deviations.
3. Divide the sum by the number of scores minus 1 (n-1).
For the given sample, square each deviation and sum up the squared deviations:
(-3)^2 + 2^2 + (-1)^2 + 1^2 + (-1)^2 + 2^2 = 9 + 4 + 1 + 1 + 1 + 4 = 20
Divide the sum by the number of scores minus 1: 20 ÷ (6 - 1) = 20 ÷ 5 = 4
The variance of the sample is 4.
Step 4: Calculate the Standard Deviation
Finally, to find the standard deviation, take the square root of the variance:
√4 = 2
Hence, the standard deviation of the sample is 2.
Summary:
The mean of the sample is 5, and the standard deviation is 2.
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.