Frequency Distributions

PSY 211

8-29-07

Frequency Distribution

• Definition: Table/graph showing number of participants in each category

x f

Basketball 23

Baseball 17

Football 29

Other 7

If I gave a 5-point quiz to a class, the distribution of scores might look like this:

X f

5 12

4 19

3 11

2 6

1 9

0 3

Find the proportion (p) of people who have each score. p = f / N

X f p

5 12 .20

4 19 .32

3 11 .18

2 6 .10

1 9 .15

0 3 .05

Find the percentage (%) of people who have each score. % = p * 100

X f p %

5 12 .20 20

4 19 .32 32

3 11 .18 18

2 6 .10 10

1 9 .15 15

0 3 .05 5

If the scores are spread out, it is okay to group them into a range of scores

Some data:

96, 68, 74, 82, 86,

51, 66, 82, 91, 85,

77,89,88,88,61,90,

79,86,80,92,88,69,

44,99,81,82

X f

91-100 5

81-90 12

71-80 3

61-70 4

51-60 1

41-50 1

Turn a frequency table into a histogram:

X f

5 12

4 19

3 11

2 6

1 9

0 3 Scores

Turn a frequency table into a histogram:

X f

5 12

4 19

3 11

2 6

1 9

0 3 Scores

f

0

5

10

15

20

0 1 2 3 4 5

Histogram of Quiz Scores

f

Scores

0

5

10

15

20

0 1 2 3 4 5

Polygon of Quiz Scores

f

Scores

Now turn that histogram (bar graph) into a polygon (shape, line graph)

6

You could graph relative frequencies (percentages) instead of frequencies

X f p %

5 12 .20 20

4 19 .32 32

3 11 .18 18

2 6 .10 10

1 9 .15 15

0 3 .05 5Scores

%

Relative Histogram of Quiz Scores

%

Scores

0

10

20

30

40

0 1 2 3 4 5

Distribution Shape

• Rather than graph an entire distribution, sometimes researchers just describe its shape and provide a few statistics

x

f

x

f

x

f

Symmetrical distributions: look the same on both sides, mirror images

x

f

x

f

*Skewed distribution: lopsided, have a tail off to one side

Tail indicates the direction of the skew (+ or -)

Positive Skew

Negative Skew

x

f

x

f

In actuality, frequency polygons look more like these

Negative Skew

Symmetrical

Percentile Rank

• Used to describe where a score falls in a given distribution

x

f

x

f

50th percentile 95th percentile

Finding Percentile Ranks

X f cf

5 12 60

4 19 48

3 11 29

2 6 18

1 9 12

0 3 3

cf = cumulative frequency, number of people with that score or lower

Finding Percentile Ranks

X f cf c%

5 12 60 100%

4 19 48 80%

3 11 29 48%

2 6 18 30%

1 9 12 20%

0 3 3 5%

c% = cumulative percentage or percentile rank, percent with that score or lower