Student Name:

Statistics Exercise

Complete the following exercises and submit for grading by the end of this week.  No statistical software is required; you should be able to execute all of the mathematical operations with a standard calculator.  Type your answers directly into this document and Save.

Suppose you have magically changed places with the professor teaching this course and that you have just administered an examination that consists of 100 multiple-choice items (where 1 point is awarded for each correct answer). The distribution of scores for the 25 students enrolled in your class could theoretically range from 0 (none correct) to 100 (all correct).   Below are your student’s scores.  You will use this raw data to complete all of the calculations in this assignment:

Student	Score (number correct)
Judy	78
Joe	67
Lee-Wu	69
Miriam	63
Valerie	85
Diane	72
Henry	92
Esperanza	67
Paula	94
Urian	69
Leroy	61
Ronald	96
Vinnie	73
Bianca	79
Martha	62
Bill	61
Homer	44
Robert	66
Michael	87
Jorge	76
Mary	83
Mousey	42
Barbara	82
John	84
Donna	51

One task at hand is to communicate the test results to your class. You want to do that in a way that will help students understand how their performance on the test compared with the performances of other students. Probably, the first step is to arrange the data by converting it from a casual listing of raw scores into something that immediately provides a little more information.

Display (in descending order) the test scores and complete the table below. (6 points)

Scores from Your Test (X)	Score f (frequency)	f(X)

(2 points each):

1. Identify the median of the frequency distribution. Median =

2. Identify the mode in the frequency distribution. Mode(s) =

3. What is the range of this frequency distribution? Range =

On average, how much does each score in this distribution vary from the mean score?

The steps for calculating the average deviation (AD) of a frequency distribution is as follows:

1. Determine the deviation scores for each score in the frequency distribution (in other words, how much does each individual score vary from the mean score?).
2. Find the sum of the deviation scores.

• Divide the sum of the deviation scores by the total number of scores to obtain the average deviation.

Complete the table below (10 points).

Scores from Your Test (X)	Score f (frequency)	Absolute Value of (X−x)

4. The sum of the absolute value of deviation scores = (2 points)
5. The total number of scores in the frequency distribution =
6. Therefore, average deviation (AD) = (2 points)

What is the standard deviation of this distribution?

The standard deviation is equal to the square root of the average squared deviations about the mean. More succintly, it is equal to the square root of the variance. So one way to calculate the standard deviation of a frequency distribution is to calculate the variance. Complete the table below as the first step in calculating the variance:

(10 points)

X	f	X−x	(X−x)2

(2 points each)

7. The sum of the squared values of deviation scores =
8. Variance = Sum of the squared values of deviation scores ÷ total number of scores
9. Therefore, variance =
10. Standard deviation = √Variance =

Think about how you will communicate this data to the class.

(2 points each)

11. What type of frequency distribution would you use?
12. Which type of graph would you use to represent the data?
13. Which measure of central tendency would you use to represent the data?
14. Which measure of variability would you use to represent the data?

It may be meaningful to your students to reference a normal curve when communicating the results.  This may be accomplished by calculating z scores and T scores.

z scores

The formula for calculating z scores is as follows:

In the equation, x is the mean of the frequency distribution and S is the standard deviation of the frequency distribution.  Complete the table by calculating the z score.

(25 points)

X	f	X−x	z = (X−x) ÷ S

T scores = 10z + 50

Complete the table by calculating the T score.

(25 points)

Score	f (frequency)	T