• The central tendency is the extent to which all the data values group around a typical or central value.

• The variation is the amount of dispersion or scattering of values

• The shape is the pattern of the distribution of values from the lowest value to the highest value.

• The arithmetic mean (often just called the “mean”) is the most common measure of central tendency

• The arithmetic mean (often just called the “mean”) is the most common measure of central tendency

• The most common measure of central tendency

• Mean = sum of values divided by the number of values

• Affected by extreme values (outliers)

• In an ordered array, the median is the “middle” number (50% above, 50% below)

• Not affected by extreme values

• The location of the median when the values are in numerical order (smallest to largest):

• If the number of values is odd, the median is the middle number

• If the number of values is even, the median is the average of the two middle numbers

• Value that occurs most often
• Not affected by extreme values
• Used for either numerical or categorical (nominal) data
• There may be no mode
• There may be several modes

Which Measure to Choose?

• The mean is generally used, unless extreme values (outliers) exist.
• The median is often used, since the median is not sensitive to extreme values. For example, median home prices may be reported for a region; it is less sensitive to outliers.
• In some situations it makes sense to report both the mean and the median.

Review Example

Summary

• Simplest measure of variation
• Difference between the largest and the smallest values:

Why The Range Can Be Misleading

• Ignores the way in which data are distributed

• Sensitive to outliers

The Sample Variance

• Low variation: more points close to the mean

• High variation: more points far from the mean

• So, measures the distance to the mean

The Sample Variance

• Average (approximately) of squared deviations of values from the mean

The Sample Standard Deviation

• Most commonly used measure of variation
• Shows variation about the mean
• Is the square root of the variance
• Has the same units as the original data

Comparing Standard Deviations

Z-Score

Z-Score

• Suppose the mean math SAT score is 490, with a standard deviation of 100.
• Compute the Z-score for a test score of 620.

• Descriptive statistics discussed previously described a sample, not the population.

• Summary measures describing a population, called parameters, are denoted with Greek letters.

• Important population parameters are the population mean, variance, and standard deviation.

The mean µ

• The population mean is the sum of the values in the population divided by the population size, N

The Variance σ2

• Average of squared deviations of values from the mean