Find the Standard Deviation for this distribution of scores. Then find the z score for X=86. N=17
98, 92, 86, 84 , 76, 74 , 72 , 72, 72 , 70, 70, 66, 60 ,54, 32,30
Although you indicate n = 17, you only have 16 scores listed.
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
Z = (score-mean)/SD
I'll let you do the calculations.
To find the standard deviation for this distribution of scores, you can follow these steps:
Step 1: Calculate the mean (average) of the scores.
Add up all the scores and then divide the sum by the number of scores (N). In this case, the sum of the scores is 1352 (98 + 92 + 86 + 84 + 76 + 74 + 72 + 72 + 72 + 70 + 70 + 66 + 60 + 54 + 32 + 30), and N is 17. So, the mean is 1352/17 = 79.53 (rounded to two decimal places).
Step 2: Find the differences between each score and the mean.
Subtract the mean from each score. For example:
Score 1: 98 - 79.53 = 18.47
Score 2: 92 - 79.53 = 12.47
Score 3: 86 - 79.53 = 6.47
Continue this for all scores.
Step 3: Square each difference.
Take each of the differences calculated in Step 2, and square them. For example:
Score 1 difference squared: 18.47^2 = 341.0809
Score 2 difference squared: 12.47^2 = 155.2009
Score 3 difference squared: 6.47^2 = 41.9209
Continue this for all differences.
Step 4: Calculate the sum of the squared differences.
Add up all the squared differences calculated in Step 3. For this example, the sum of the squared differences is 5020.2591 (rounded to four decimal places).
Step 5: Divide the sum of the squared differences by N-1.
Divide the sum calculated in Step 4 by N-1 (where N is the number of scores). In this case, N is 17, so 5020.2591/16 = 313.7662 (rounded to four decimal places).
Step 6: Take the square root of the result from Step 5.
Calculate the square root of the result from Step 5. In this case, the square root of 313.7662 is approximately 17.7163 (rounded to four decimal places).
Therefore, the standard deviation for this distribution of scores is approximately 17.7163 (rounded to four decimal places).
To find the z-score for X=86:
Step 1: Subtract the mean from the value.
In this case, the mean is 79.53 (rounded to two decimal places), and X=86. So, 86 - 79.53 = 6.47.
Step 2: Divide the result obtained in Step 1 by the standard deviation.
The standard deviation is approximately 17.7163 (rounded to four decimal places). So, 6.47 / 17.7163 = 0.3655 (rounded to four decimal places).
Therefore, the z-score for X=86 is approximately 0.3655 (rounded to four decimal places).