Measures of Central Tendency

Topics Covered in this Session

Measures of Central Tendency

Definition – averages or what is typical for a group of values such as scores, grades, etc. The three major measures of central tendency are the mean, median and mode.

Mean - is the arithmetic average. For example, the mean of the following group of numbers 60,70,80,90,100 is 80 and is calculated by dividing the sum of the values by the number of values in the group (N=5): 60+70+80+90+100 = 400/5 = 80
The mean is the most commonly used statistical measure and is most useful in depicting what is typical for a group of values. It is also used extensively with other statistical formulae.
Median - is the mid-point in a group of values, above and below which one-half of the values fall. In the above example (60,70,80,90,100), 80 is the median with two scores above and two scores below 80. If there are an even number of values, the median is the point halfway between the two middle values. For example, in the following group of values 70,70, 80, 82, 90, 100; the median is 81 (halfway between the two middle values 80 and 82).

The median is useful when very high or very low numbers may distort or skew a mean. For example, the mean ($110,400.) for the following annual salaries

$ 10000.
$ 12000.
$ 15000.
$ 15000.
$ 500000.

$552000. / 5 = $110400. – Calculation for the Mean
is skewed because of one very high salary ($500000.) and does not accurately indicate what is typical of this group. The median which is $15000. is more typical and a more accurate measure of central tendency in this example. The median is used extensively when presenting data such as salaries, incomes, prices of houses, etc.

Of the above measures of central tendency, the mean is by far the most extensively used in educational research.

T-test

The t-test is a parametric (assumes normal distribution) test to determine the significance of the difference between the mtheans of two groups of values. The t-test uses the mean, the variance and a Table of Critical Values for a “t” Distribution to calculate a t value. The rejection or acceptance of the significance of the differences in two means is based on a standard that no more than 5% of the difference is due to chance or sampling error, and that the same difference would occur 95% of the time should the test be repeated. Some researchers use a more rigorous standard of 1% (.01 Level), and that the same difference would occur 99% of the time should the test be repeated.

The t-test usually is displayed in a study or report as follows: The experiment or treatment group (M=86.50, SD=4.31) scored significantly higher than the control group (M=79.10, SD=5.22), t(80) = 4.90, p<.05 where

M = Mean
SD = Standard Deviation
t = t value
Number in parenthesis (80) after the t value = N (number of cases adjusted for degrees of freedom)
p = indicates the level of statistically significant difference (i.e .05 level) between the two means.

In the above example, p is the bottom line value and indicates at what level a statistically significant difference exists.

Analysis of Variance (ANOVA)

Analysis of variance is a statistical measure used for determining whether differences exist among two or more groups. It does this by comparing the means of the groups to see if they are statistically different. Analysis of variance uses the mean, the variance and a Table of Critical Values for “F” Distribution to calculate an F statistic. Analysis of variance is a parametric (assumes normal distribution) test. Statistical significance of the differences in two or more means is based on a standard that no more than 5% (.05 Level) of the difference is due to chance or sampling error, and that the same difference would occur 95% of the time should the test be repeated. Some researchers use a more rigorous standard of 1% (.01 Level), and that the same difference would occur 99% of the time should the test be repeated.

Analysis of variance can be used for several different types of analyses.
Oneway Analysis of Variance - assumes there are two variables with one variable a dependent, interval or ratio variable (numerical data that show quantity and direction), and one variable, an independent, nominal variable or factor such as an ethnicity code or sex code.
N-way Analysis of Variance - assumes there are more than two variables with one variable a dependent, interval or ratio variable and two or more, independent, nominal variables or factors such as ethnicity code or sex code.
Multiple Regression - assumes there are more than two variables with one variable a dependent, interval or ratio variable and two or more, independent, interval or ratio variables such as test scores, income, grade point average, etc.
Analysis of Covariance - assumes there are more than two variables with one variable a dependent, interval or variable and two or more variables are a combination of independent, nominal, interval or ratio variables

Depending on the options used, ANOVA can be displayed in different ways in a study or a report. For an N-way ANOVA, the following is typical. The analysis of variance indicated that there were significant differences among the four groups F(3, 96)=7.50, p<.01 where

F = the F statistic
The two numbers in parentheses (3,96) = the number of groups and N (number of cases adjusted for degrees of freedom)
p = indicates the level of statistically significant difference (i.e .01 level) among the meansIn the above example, p is the bottom line value and indicates at what level a statistically significant difference exists.

Scheffe Test

The Scheffe test is used with ANOVA (Analysis of Variance) to determine which variable(s) among several independent variables is statistically the most different.

Chi-Square

T-test and analysis of variance are parametric statistical procedures that assume that the distributions are normal or nearly normal and is used when variables are continuous such as test scores and grade point averages. Chi-square is a nonparametric statistical procedure used to determine the significance of the difference between groups when data are nominal and placed in categories such as gender or ethnicity. This procedure compares what is observed against what was expected.

FOR MORE INFORMATION ON THE TOPICS COVERED IN THIS SESSION, PLEASE REFER TO THE APPENDIX IN A.G. PICCIANO "EDUCATIONAL RESEARCH PRIMER" AS WELL AS THE MANUALS AND DOCUMENTATION PROVIDED BY SPSS, INC.