Chapter 2

CREATING AND USING FREQUENCY DISTRIBUTIONS

Going Forward

Your goals in this chapter are to learn:• What frequency is and how a frequency

distribution is created• When to graph frequency distributions using a

bar graph, histogram, or polygon• What normal, skewed, and bimodal

distributions are• What relative frequency and percentile are and

how we use the area under the normal curve to compute them

New Symbols and Terminology

• Raw scores are the scores we initially measure in a study

• The number of times a score occurs in a set of data is the score’s frequency

• A frequency distribution organizes the scores based on each score’s frequency

New Symbols and Terminology

• The frequency of a score is symbolized by f• N is the total number of scores in the data

Understanding Frequency Distributions

Frequency Distribution

• A frequency distribution table shows the number of times each score occurs in a set of data

• N is the total of all the individual frequencies in the f column of a frequency distribution table

Raw Scores

Use the following raw scores to construct a frequency distribution table.

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Frequency Distribution Table

Graphing Frequency Distributions

• A frequency distribution graph always shows the scores on the X axis and their frequency on the Y axis

• The type of measurement scale (nominal, ordinal, interval, or ratio) determines whether we use– A bar graph– A histogram– A polygon

Frequency Bar Graph for Nominal and Ordinal Data

Histogram for a Small Number of Different Interval or Ratio Scores

Frequency Polygon for Many Different Interval or Ratio Scores

Types of Frequency Distributions

The Normal Distribution

• A bell-shaped curve

• Called a normal curve or a normal distribution

• Symmetrical

• The far left and right portions containing the relatively low-frequency, extreme high or low scores are called the tails of the distribution

An Ideal Normal Distribution

Skewed Distributions

• A skewed distribution is not symmetrical as it has only one pronounced tail

• A distribution may be either negatively skewed or positively skewed

• The direction in which the distinctive tail slopes indicates whether the skew is positive or negative

Negatively Skewed Distribution

A negativelyskewed distribution contains extremelow scores having low frequency, butdoes not contain low-frequency, extreme high scores.

Positively Skewed Distribution

A positivelyskewed distributioncontains extremehigh scores having low frequency, butdoes not contain low-frequency, extreme low scores.

Bimodal Distribution

A bimodaldistribution is asymmetricaldistributioncontaining twodistinct humps.

Frequency Distribution Shape

• The shape of the frequency distribution is an important characteristic of the data

• The shape also determines which statistical procedures we should use

Relative Frequency and the Normal Curve

Relative Frequency

• Relative frequency is the proportion of the time a score occurs in a sample

• The formula for computing a score’s relative frequency is

Relative frequency =N

f

Finding Relative Frequency Using the Normal Curve

The proportion of the total area under the normal curve occupied by a group of scores corresponds to the relative frequency of those scores.

Understanding Percentile and Cumulative Frequency

Percentile

• A percentile is the percent of all scores in the data located below a score

• One way to determine a score’s percentile is to find the area under the normal curve to the left of the score

Cumulative Frequency

The cumulative frequency is the number of

scores in the data that are at or below a

particular score.

Percentiles

Normal distribution showing the area under the curve to the left of selected scores.

Example

Using the following data set, find the relative frequency and cumulative frequency of the score 12.

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example

The frequency table for this set of data.

Example—Relative Frequency

• The frequency for the score of 12 is 1 and N = 18

• Therefore, the relative frequency of 12 is

06.018

1

N

fRelative Frequency

Example—Cumulative Frequency

• There is one score at 12 and two scores below 12 (one score of 11 and one score of 10)

• Therefore, the cumulative frequency of 12 is 3