Chapter 3

SUMMARIZING SCORES WITH MEASURES OF CENTRAL TENDENCY

Moving Forward

Your goals in this chapter are to learn:• What central tendency is• What the mean, median, and mode indicate

and when each is appropriate• The uses of the mean• What deviations around the mean are• How to interpret and graph the results of an

experiment

New Symbols and Procedures

• The symbol is the Greek letter “S” and is called sigma.

• This symbol means to sum (add).

• You will see it used with a symbol for scores such as X. This is pronounced as the “sum of X” and means to find the sum of the X scores.

What is Central Tendency?

Central Tendency

• Measures of central tendency answer the question: – “Are the scores generally high scores or generally

low scores?”

What is Central Tendency?

• A measure of central tendency is a statistic summarizing the location of a distribution on a variable

• It indicates where the center of the distribution tends to be located

Computing the Mean, Median,and Mode

The Mode

• The mode is the score having the highest frequency in the data

• The mode is used to describe central tendency when the scores reflect a nominal scale of measurement

Unimodal Distributions

When a polygonhas one hump (such as on the normal curve) thedistribution is called unimodal.

Bimodal Distributions

When a distributionhas two scorestied for the mostfrequently occurringscore, it is calledbimodal.

The Median

• The median is the score at the 50th percentile

• The median is used to summarize ordinal or highly skewed interval or ratio scores

Determining the Median

• When data are normally distributed, the median is the same score as the mode

• The symbol for the median is Mdn• Use the median for ordinal data or when you

have interval or ratio scores in a very skewed distribution

Determining the Median

When data are normally distributed, the median is the same score as the mode. When data are only approximately normally distributed:

1. Arrange the scores from lowest to highest2. Determine N3. If N is an odd number, the median is the score in

the middle position4. If N is an even number, the median is the

average of the two scores in the middle

The Mean

• The mean is the score located at the exact mathematical center of a distribution

• The mean is used to summarize interval or ratio data in situations when the distribution is symmetrical and unimodal

Computing a Sample Mean

• The formula for the sample mean is

N

XX

Comparing the Mean,Median, and Mode

• All three measures of central tendency are located at the same score on a perfectly normal distribution

• In a roughly normal distribution, the mean, median, and mode will be close to the same score

• The mean inaccurately describes a skewed distribution

Central Tendency andSkewed Distributions

Applying the Mean to Research

Deviations

• A score’s deviation is equal to the score minus the mean

• In symbols, this is • The sum of the deviations around the mean

is the sum of all the differences between the scores and the mean

• It is symbolized by

XX

XX

Summarizing Research

• Scores are from the dependent variable• Choose the mean, median, or mode based on

1. The scale of measurement used on the dependent variable and

2. The shape of the distribution, if you have interval or ratio scores

Graphing the Resultsof an Experiment

• Plot the independent variable on the X (horizontal) axis and the dependent variable on the Y (vertical) axis

• Create a line graph when the independent variable is an interval or a ratio variable

• Create a bar graph when the independent variable is a nominal or ordinal variable

Line GraphsA line graph uses straight lines to

connect adjacent data points

Bar Graphs

The bar above each condition on the X axis is placed to the height on the Y axis that correspondsto the mean score for that condition.

Example

For the following data set, find the mode, the median, and the mean.

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example—Mode

• The mode is the most frequently occurring score

• In this data set, the mode is 14 with a frequency of 6

Example—Median

The median is the score at the 50th percentile. To find it, we must first place the scores in order from smallest to largest.

10 11 12 13 13 13

13 14 14 14 14 14

14 15 15 15 15 17

Example—Median

• Since this data set has 18 observations, the median will be half-way between the 9th and 10th score in the ordered dataset

• The 9th score is 14 and the 10th score also is14. To find the midpoint, we use the formula:

• The median is 14

142

1414

Example—Mean

• For the mean, we need X and N • We know N = 18

67.1318

246

246

N

XX

X