Measures of Central Tendency
The mean of raw data is the sum divided by the count. For grouped data, the mean is the sum of frequency times value divided by the sum of frequencies. Adding k to every value adds k to the mean, and multiplying every value by k multiplies the mean by k.
The median is the middle value of ordered data. If the count is odd, it is the value at position n plus 1 over 2. If even, it is the average of the two middle values. The mode is the value that occurs most often. For roughly symmetric data, mode equals 3 times median minus 2 times mean.
Measures of Spread
The range is the largest value minus the smallest. The variance is the average of the squared deviations from the mean. The standard deviation is the square root of the variance. The coefficient of variation is the standard deviation over the mean times 100 percent. Adding k to every value leaves the standard deviation unchanged, and multiplying every value by k multiplies it by k.
Frequency Distributions
The class width is the upper limit minus the lower limit. The class mid-value is half the sum of the limits. Cumulative frequency is the running total of frequencies.
Shortcuts
Shifting the data does not change the spread; scaling the data scales the spread. For the mean, guess an assumed mean, average the deviations, and add back the guess. Sort data first for median problems. If all values are the same, the standard deviation is zero.
Worked Examples
Example 1: Find the mean, median and range of 5, 7, 8, 9, 11.
Solution: The mean is 8, the median is 8 and the range is 6.
Example 2: Find the mode if the median is 20 and the mean is 22.
Solution: Three times 20 minus twice 22 equals 16.
Example 3: Five students average 60. A sixth scores 90. Find the new average.
Solution: The new total is 390, so the average is 65.
Quick Check
Question 1: What is the median of an even-sized data set?
Answer: The average of the two middle values.
Question 2: Does adding a constant change the standard deviation?
Answer: No.
Question 3: What is the empirical relation between mean, median and mode?
Answer: Mode equals 3 times median minus 2 times mean.