Descriptive Statistics - Measures of Central Tendency & Dispersion

Standard Deviation [(J) pp. 375-376, (L) pp. 149-150]

A very precise measure of dispersion. Appropriate only to interval &/or ratio-level data. Square root of variance; expressed in the data's original units. Expresses the average deviation from the mean for observations of a single variable.

3 Measures of Central Tendency

All 3 express the "center of gravity" of the observations for a single variable; useful in summarizing its essential characteristics.

Measures of Dispersion, Variability or Spread

Express the degree of variation within a variable. Useful for describing & comparing variables.

Measures of Dispersion

Range—basic measure of dispersion, appropriate to ordinal & ratio-level data. Highest value of a variable minus the lowest value. Large range = much variation. Inter-quartile range--the middle 50 % of observations. Not so sensitive to extreme cases.

Calculating "Z" Scores

Subtract the mean of the variable from the given score. Divide that value by the standard deviation of the data set. The resulting number ("z" score) expresses how many standard deviations from the mean is a given score.

Standardized or "Z" scores( L p.152)

Technique applies to normally distributed, ratio-level variables. Expresses deviation from the mean in terms of standard deviations. Useful for comparing scores within a data set. Measures how typical is a given score as compared to the group of scores. Simple to calculate & easy to interpret.

Mode or modal observation

The value occurring most often in a data set. Can be used with all 4 levels of measurement. The only measure of central tendency appropriate to nominal level data.

Median

The value of a variable having 50% of the observations below it and 50% above it. Can be used with ordinal, interval &/or ratio-level data. Esp. useful for data sets having extreme values.

Mean (arithmetic average)

Total of all the observations of a variable divided by the number of cases. Most precise measure of central tendency. Appropriate only for interval &/or ratio-level variables and data.

Standard Deviation (cont.)

Useful for comparing variables or data sets for degree of variation within them. Higher standard deviation = more variation within the variable. 68% of observations fall within one standard deviation of the mean for a normally distributed variable; 95% within two standard deviations.