Chapter 3Frequency Distributions

Frequency Distributions

• A frequency distribution lists the number of times, or frequency (f), each raw score (X) in a set of score (distribution) appears.

X f

3 2

2 3

1 2

Ex: 1, 1, 2, 2, 2, 3, 3

It’s easy to spot the mode.

Constructing a Frequency Distribution

14, 14, 10, 8, 14, 16, 8, 10, 10, 8, 10, 16, 16, 6, 20, 16, 9, 10, 17, 10, 18, 14, 14, 8

Step 1: Find the highest and lowest scores.

Step 2: Construct a table including all numbers. By convention, the highest score is listed first and the lowest score is listed last.

Constructing a Frequency Distribution

Step 3: Make a tally mark for each score.

Step 4: Count up the tally marks and fill in the frequency column.

Frequency Distributions

Like inferential statistics, descriptive statistics, and graphs, frequency distributions are judgment aids. They make it easier for a scientist or another person to react meaningfully to the results of an experiment or to other information (data).

Grouped frequency distribution

In a grouped frequency distribution, the total range of scores is divided into a number of mutually exclusive (nonoverlapping) and collectively exhaustive categories (class intervals) and the frequency of scores in each category is reported.– Categories typically are of the same width (e.g., 1-4, 5-9, 10-

14).– The number of categories depends on the data of interest; the

textbook recommends 10-15 categories.– To determine the size of class intervals, subtract the lowest

score from the highest score and divide by the number of desired class intervals.

Six Groups

Eleven Groups

Sixteen Groups

Your Turn: Activity #1

17, 3, 16, 16, 28, 5, 3, 23, 31, 19, 2, 20, 19, 17, 27, 25, 18, 15, 19, 6, 22, 9, 15, 22, 1, 18, 16

1. Make 10 categories for your frequency distribution.

2. Create a frequency distribution using the above data.

3. Put your name on your paper.

Discrete vs. Continuous Variables

• Discrete variables can take only specific values (usually whole numbers).– Ex: The number of people in a family can be

three, but it can’t be 3.45.

• Continuous variables can take an infinite number of values. – Ex: The amount of alcohol consumed can

take on any value. For example, someone could consume 29.5698359167 oz. of tequila.

Apparent vs. Real LimitsX f

8

7

6

5

• Apparent Limits: When we are working with continuous variables, this table won’t do. It has gaps in it. (In fact, the probability of getting a value of exactly 6 is infinitely small.)

• Real Limits: We can transform this table into one that includes all possible values in the range.

• In this case, we will use the original value ± .5 to eliminate the gaps.

X f

7.5 – 8.5

6.5 – 7.5

5.5 – 6.5

4.5 – 5.5

Percentage (or Relative) Distributions

It is possible to express frequencies as percentages, which may make frequency distributions more useful as judgment aids.

f = frequency (of a particular score)N = total number of scores

Why is it Useful?

• One of the main advantages is that you can compare results from samples of different sizes.

Activity #2• Write on a small piece of paper the number of times you think

most university students have had sexual intercourse in the last 30 days.

• Pass the paper to the end of your row.

• Our categories: 0, 1, 2-5, 6-10, 11+

• Change the number in each category to a percentage and construct a frequency table

• Put your name on the paper.

How many times in the last 30 days have you had sexual intercourse?

0 1 2-5 6-10 11+0%

10%

20%

30%

40%

50%

60%

MaleFemaleTotal

How many times in the last 30 days have most students had sexual intercourse?

0 1 2-5 6-10 11+0%

10%

20%

30%

40%

50%

60%

MaleFemaleTotal

(from Wechsler & Kuo, 2000)

Cumulative Frequency Distributions

• Starting with the lowest score and moving to the highest, the frequency is accumulated.

• For example: X

3

2

1

f

2

3

2

Cum f

7

5

2

Why is it useful?

• The Cumulative Frequency Distribution allows you to see the median and the distribution of scores around the median.

Cumulative Percentage Distributions

• Cumulative percentage frequencies also can easily be calculated.

• Use the data from the previous chart and this formula:

• You try.X

3

2

1

f

2

3

2

Cum f

7

5

2

Cum %age

100

71

29

Why is it Useful?

• The Cumulative Percentage Distribution allows you to see the percentage of scores that fall at or below a given score.

Students’ Estimates of Binge Drinkers in their School

RESPONSE (%) %age Cum %age

0-9 8.2 8.2

10-19 11.7 19.8

20-29 14.4 34.2

30-39 16 50.2

40-49 9.2 59.4

50-59 10.3 69.7

60-69 9.7 79.4

70-79 6.7 86.1

80-100 3.2 89.3

Don’t know 10.7 100.0

Combine All Methods

X

3

2

1

f

2

3

2

Cum f

7

5

2

Cum %age

100

71

29

%age

29

43

29

Homework

• Prepare for Quiz 3 over Chapter 3

• Read Chapter 4

• Finish Chapter 3 Homework (check course website or WebCT)

• Prepare for Basic Math Re-test