a) What is the standard deviation of the set of data?
b) What raw score will have 15% of the data set above it?
c) If I have a raw score of 21 what percentage of the scores, are below my score?
16, 16, 89, 76, 24, 45, 98, 20, 22, 28, 34, 35, 76, 67, 48, 37
a) 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
b) Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.15) and its Z score.
Insert data in above equation to get the raw score.
c) Use same equation and table.
a) 27.08
a) To find the standard deviation of a data set, you can use the formula that involves calculating the variance first and then taking the square root of the variance. The steps to find the standard deviation are as follows:
1. Find the mean (average) of the data set by adding all the numbers together and dividing by the total count of numbers. In this case, the mean is (16+16+89+76+24+45+98+20+22+28+34+35+76+67+48+37) / 16 = 51.8125.
2. Subtract the mean from each individual data point, square the result, and add up all the squared differences. For example, for the first data point 16, the squared difference is (16 - 51.8125)² = 1353.90625. Do this for all data points and add the results together. The sum of squared differences is 15618.6875.
3. Divide the sum of squared differences by the total count of data points to calculate the variance. In this case, the variance is 15618.6875 / 16 = 976.167969.
4. Finally, take the square root of the variance to find the standard deviation. So, the standard deviation of the given data set is √976.167969 ≈ 31.236.
b) To find a raw score that will have 15% of the data set above it, you need to use the cumulative distribution function (CDF) of the data set. Start by arranging the data in ascending order:
16, 16, 20, 22, 24, 28, 34, 35, 37, 45, 48, 67, 76, 76, 89, 98
Next, calculate the percentage rank (percentile) for each raw score. In this case, the percentage rank for each raw score can be found by dividing its position in the ordered data set (from 1 to 16) by the total count of data points (16) and multiplying by 100.
For example, for the raw score 16, the calculation is (1 / 16) * 100 = 6.25%. Do this for all raw scores.
Once you have the percentage ranks, find the raw score that corresponds to a 15% cumulative percentage. In this case, look for the raw score where the cumulative percentage is closest to 15%. Based on the calculations, the raw score with the closest cumulative percentage to 15% is 34.
c) To find the percentage of scores that are below a certain raw score, you can use a similar approach as in part b, but this time you are looking for the cumulative percentage above the raw score. Follow the steps below:
1. Arrange the data in ascending order as shown in part b: 16, 16, 20, 22, 24, 28, 34, 35, 37, 45, 48, 67, 76, 76, 89, 98.
2. Calculate the percentage rank (percentile) for each raw score by dividing its position in the ordered data set by the total count of data points (16) and multiplying by 100.
3. Find the raw score that corresponds to a cumulative percentage above 21. In this case, look for the raw score where the cumulative percentage is closest to or just above the desired percentage. Based on the calculations, the cumulative percentage for the raw score of 21 is 25%, which means that 25% of the scores are below 21.