hi, how would i calculate the standard deviation of the following sampling distribution of means. apparently this is the longhanded way of calculating standard error.
*a population of scores contains exactly 5 scores : 2, 3, 6, 8, 11
(2, 2) ... 2
(2, 3) ... 2.5
(2, 6) ... 4
(2, 8) ... 5
(2, 11) ... 6.5
(3, 2) ... 2.5
(3, 3) ... 3
(3, 6) ... 4.5
(3, 8) ... 5.5
(3, 11) ... 7
(6, 2) ... 4
(6, 3) ... 4,5
(6, 6) ... 6
(6, 8) ... 7
(6, 11) ... 8.5
(8, 2) ... 5
(8, 3) ... 5,5
(8, 6) ... 7
(8, 8) ... 8
(8, 11) ... 9.5
(11, 2) ... 6,5
(11, 3) ... 7
(11, 6) ... 8,5
(11, 8) ... 9,5
(11, 11) ... 11
To calculate the standard deviation of the sampling distribution of means, you need to follow these steps:
1. Calculate the mean of the population. In this case, the population contains five scores: 2, 3, 6, 8, and 11. To find the mean, sum up all the scores and divide by the total number of scores. In this case, the sum is 30, and since there are five scores, the mean is 30/5 = 6.
2. For each sample mean in the sampling distribution, subtract the population mean and square the result. For example, for the first sample mean of (2, 2), the difference from the population mean is 2 - 6 = -4, and squaring it results in 16.
3. Repeat step 2 for all sample means in the sampling distribution and calculate the sum of the squared differences from the population mean. In this case, the sum of the squared differences adds up to 111.
4. Divide the sum of the squared differences from step 3 by the total number of samples in the sampling distribution. In this case, there are 25 sample means, so the calculation is 111/25 = 4.44.
5. Lastly, take the square root of the result from step 4 to obtain the standard deviation of the sampling distribution of means. The square root of 4.44 is approximately 2.11.
Therefore, the standard deviation of the sampling distribution of means for this population of scores is approximately 2.11.
To calculate the standard deviation of the sampling distribution of means, you will need to follow these steps:
Step 1: Calculate the mean of the population scores.
Add up all the scores in the population and divide the sum by the total number of scores.
Mean = (2 + 3 + 6 + 8 + 11) / 5 = 30 / 5 = 6
Step 2: Calculate the squared deviation for each score in the sampling distribution.
For each pair of scores in the sampling distribution, subtract the mean (6) from the score and square the result.
(2, 2) --> (2 - 6)^2 = 16
(2, 3) --> (3 - 6)^2 = 9
(2, 6) --> (6 - 6)^2 = 0
(2, 8) --> (8 - 6)^2 = 4
(2, 11) --> (11 - 6)^2 = 25
Continue this process for all the pairs of scores in the sampling distribution.
Step 3: Calculate the sum of squared deviations.
Add up all the squared deviations calculated in the previous step.
Sum of Squared Deviations = 16 + 9 + 0 + 4 + 25 + 9 + 0 + 1 + 4 + 25 + 0 + 2.25 + 0.25 + 4 + 23.25 + 1 + 0.25 + 1 + 4 + 20.25 + 0.25 + 1 + 20.25 + 9 + 0 = 191.25
Step 4: Calculate the variance.
Divide the sum of squared deviations by the total number of pairs of scores in the sampling distribution (25 in this case).
Variance = Sum of Squared Deviations / Number of Pairs = 191.25 / 25 = 7.65
Step 5: Calculate the standard deviation.
Take the square root of the variance calculated in the previous step.
Standard Deviation = √(Variance) = √(7.65) ≈ 2.77
So, the standard deviation of the sampling distribution of means is approximately 2.77.