Measures of Position

CO-4: Distinguish among different measurement scales, choose the appropriate descriptive and inferential statistical methods based on these distinctions, and interpret the results.
LO 4.4: Using appropriate graphical displays and/or numerical measures, describe the distribution of a quantitative variable in context: a) describe the overall pattern, b) describe striking deviations from the pattern
LO 4.14: Define and interpret measures of position (percentiles, quartiles, the five-number summary, z-scores).
Video: Measures of Position (2 videos, 4:20 total)

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Although not a required aspect of describing distributions of one quantitative variable, we are often interested in where a particular value falls in the distribution. Is the value unusually low or high or about what we would expect?

Answers to these questions rely on measures of position (or location). These measures give information about the distribution but also give information about how individual values relate to the overall distribution.


A common measure of position is the percentile. Although there are some mathematical considerations involved with calculating percentiles which we will not discuss, you should have a basic understanding of their interpretation.

In general the P-th percentile can be interpreted as a location in the data for which approximately P% of the other values in the distribution fall below the P-th percentile and (100 –P)% fall above the P-th percentile.

The quartiles Q1 and Q3 are special cases of percentiles and thus are measures of position.

Five-Number Summary

The combination of the five numbers (min, Q1, M, Q3, Max) is called the five number summary, and provides a quick numerical description of both the center and spread of a distribution.

Each of the values represents a measure of position in the dataset.

The min and max providing the boundaries and the quartiles and median providing information about the 25th, 50th, and 75th percentiles.

Standardized Scores (Z-Scores)

Standardized scores, also called z-scores use the mean and standard deviation as the primary measures of center and spread and are therefore most useful when the mean and standard deviation are appropriate, i.e. when the distribution is reasonably symmetric with no extreme outliers.

For any individual, the z-score tells us how many standard deviations the raw score for that individual deviates from the mean and in what direction. A positive z-score indicates the individual is above average and a negative z-score indicates the individual is below average.

To calculate a z-score, we take the individual value and subtract the mean and then divide this difference by the standard deviation.

gives the z-score formula in which we take the specified value, x, and subtract the sample mean, x-bar, then divide that result by the standard deviation to get the z-score

Measures of Position

Measures of position also allow us to compare values from different distributions. For example, we can present the percentiles or z-scores of an individual’s height and weight. These two measures together would provide a better picture of how the individual fits in the overall population than either would alone.

Although measures of position are not stressed in this course as much as measures of center and spread, we have seen and will see many measures of position used in various aspects of examining the distribution of one variable and it is good to recognize them as measures of position when they appear.