PPA 415 – Research Methods in Public Administration Lecture 4 – Measures of Dispersion

Introduction By themselves, measures of central tendency cannot summarize data completely. For a full description of a distribution of scores, measures of central tendency must be paired with measures of dispersion. Measures of dispersion assess the variability of the data. This is true even if the distributions being compared have the same measures of central tendency.

Introduction – Example, JCHA 1999

Introduction Measures of dispersion discussed. Index of qualitative variation (IQV). The range and interquartile range. Standard deviation and variance.

Index of Qualitative Variation Used primarily for nominal variables, but can be used with any variable with a frequency distribution. Ratio of amount of variation actually observed in a distribution of scores to the maximum variation that could exist in that distribution.

Index of Qualitative Variation Maximum variation in a frequency distribution occurs when all cases are evenly distributed across all categories. The measure gives you information on how homogeneous or heterogeneous a distribution is.

Index of Qualitative Variation

Range and Interquartile Range Range: the distance between the highest and lowest scores. Only uses two scores. Can be misleading if there are extreme values. Interquartile range: Only examines the middle 50% of the distribution. Formally, it is the difference between the value at the 75% percentile minus the value at the 25 th percentile.

Range and Interquartile Range Problems: only based on two scores. Ignores remaining cases in the distribution.

Range and Interquartile Range: JCHA 1999 Example

The Standard Deviation The basic limitation of both the range and the IQR is their failure to use all the scores in the distribution A good measure of dispersion should Use all the scores in the distribution. Describe the average or typical deviation of the scores. Increase in value as the distribution of scores becomes more heterogeneous.

The Standard Deviation One way to do this is to start with the distances between every point and some central value like the mean. The distances between the scores are the mean (Xi-Mean X) are called deviation scores. The greater the variability, the greater the deviation score.

The Standard Deviation One course of action is to sum the deviations and divide by the number of cases, but the sum of the deviations is always equal to zero. The next solution is to make all deviations positive. Absolute value – average deviation. Squared deviations – standard deviation.

Average and Population Standard Deviation

Sample Variance and Standard Deviation

Computational Variance and Standard Deviation - Sample

Examples – JCHA 1999

Examples – Average and Standard Deviation

Grouped Standard Deviation

Grouped Standard Deviation Example