# Survey Statistics

Measures of Central Tendency are generally located near the center of the distribution of values. Distribution Statistics are used to summarize the distribution of values. Measures of Variation indicates the distribution of the values.

Mean
The arithmetic mean is also called the average. The mean is the sum of all values in the set of numbers divided by the number of elements in the set.

Median
The median is simply the number in the middle of a complete set of numbers ordered from highest to lowest. If the complete ordered set of numbers contains an even number of values, the median is the average of the two middle values. The median is located at the 50th percentile. It represents the middle of the distribution of values. One-half of the values are greater than the median, and one-half of the values are less than the median.

Mode
The mode is the most frequently occurring, or repeating, value in the set of numbers.

Minimum
The minimum is the smallest number in the complete set of values.

Maximum
The maximum is the largest number in the complete set of values.

Range
The range is the difference between the maximum (largest) value and the minimum (smallest) value in the distribution of values. A large range would indicate the values are spread out.

Variance
Two sets of numbers could have the same range, mean, and median. Variability measures how much the data values are spread out from the center. Put another way, variability describes how much the values differ from each other. If most of the values are grouped near the median, the distribution has a low variability. If the values are not grouped near the median, (spread out over the range), the distribution has a high variability. Variance is the sum of the squared deviations from the mean divided by the number of elements less one.

Standard Deviation
Statisticians use the standard deviation to calculate variability. The standard deviation represents a 'typical' deviation of the values from the average. A large standard deviation indicates a high degree of variability of the data.

Deviation is defined as the distance of the measurements away from the mean. Variance is the sum of the squared deviations from the mean divided by the number of elements less one. The standard deviation is the positive square root of the variance.

Skew
In a distribution with perfect symmetry, the mean, median, and mode are the same value. However, most of the time, the distribution will be skewed away from the median. Skew is the measure of the lack of symmetry in the distribution of the data values. A positive skew means most of the data falls to the left of the mean (lower values), and the "tail" of the distribution is to the right (higher values). A negative skew indicates most of the data falls to the left of the mean, and the tail of the distribution is to the left. Kurtosis measures the heaviness of the tails in the distribution.

Quartiles
The (lower) first quartile represents the value such that 25% of the other values are below that value. The (upper) third quartile represents the value such that 25% of the other values are above that value. The Inter-quartile range is the set of values between the first and third quartiles, or the middle 50% of all values.

Page last updated: January 26, 2008