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Basic Statistics & Probability

Basic Statistics & Probability

At a Glance - Mean, Median, Mode & Range

The most commonly used statistic is the average, or finding where the middle of the data lies. There are three ways to measure the average: the mean, median, and mode

Why three ways? Good question. Each will give you a different way of looking at the numbers; depending on the question you're trying to answer (or the argument you're trying to make), any of the three could prove the most useful.

The mean is the most commonly used measure of finding the average. (In fact, in everyday language, people often use the word "average" simply to mean, um, "mean").  Finding the mean is simple: just add up all the numbers in a data set and divide by the number of data entries.

mean: sum of the data/# of the data

The median is the middle number in a data set. However, the data must be in numerical order (least to greatest or greatest to least) before finding this average. If the middle number lies between two numbers, find the mean of those two numbers (add them together and divide by 2).

median: the middle # in the data set

The mode is probably the least common way of finding the average, and in most cases is the least useful. To find the mode, just look for the number that occurs the most. There can be more than one mode, or none at all.

mode: the # that occurs the most

Finally there is the range. The range is NOT a measure of the average; however, it is often taught along with averages because it's another helpful way to measure a set of data. The range measures the "spread" of the data, how far apart the smallest and largest values are. To find the range, subtract the smallest value in the data set from the largest.

range: the largest value - the smallest value

This will all make a lot more sense with some real examples.  For our social networking survey, we found the mean, median, and range for the amount of time each group spends on social networking sites per day.

Time Spent Social Networking

As we predicted, on average girls spend at least one hour a day more on these sites than boys. Both the mean and median for the girls was significantly higher.

This Week on Prancing with the B-List Celebrities

Here are the contestants' scores on this week's episode of our favorite show, Prancing with the B-List Celebrities:

Evan L52
Nicole S50
Pamela A47
Chad O44
Erin A39
Jake P38
Niecy N36
Kate G32

Now, let's find the three averages and the range for the contestants' scores.

StatHow to FindExplanation
338/8 = 42.25
Add all the scores and divide by 8, the number of contestants. The mean is 42.25.
Median52 50 47 44 (red arrow) 39 38 36 32
(44+39)/2 = 83/2 = 41.5
First put the scores in order, then find the middle value. In this set, the middle value lies between 44 and 39, so add these middle numbers together and divide by 2. The median is 41.5.
ModeNo modeNo score occurs more than once, so there is no mode for this data set.
Range52-32=20Subtract the smallest score from the largest. The range is 20 points.

For this data set, there are really only two measures of the average, since there is not a mode. Both the mean and median could be used to describe the average. If you were Evan, would you rather call out the mean or the median? What if you were Kate?

Look Out: the range of a data set does NOT measure the average of the data.

Example 1

These are the scores from last week's geometry test:

90, 94, 53, 68, 79, 84, 87, 72, 70, 69, 65, 89, 85, 83, 72

You earned a score of 72. Your mom asks you how you did on the test compared to the rest of the class. Calculate the three measures of the average, and decide what to tell your mom.

Example 2

These are the top ten final scores for the combined results of the Ladies' Figure Skating event at the 2010 Winter Olympics:

Figure Skating
Yu-Na Kim228.56
Mao Asada205.50
Joanie Rochette202.64
Mirai Nagasu190.15
Miki Ando188.86
Laura Lepisto187.97
Rahail Flatt182.49
Akiko Suzuki181.44
Alena Leonova172.46
Ksenia Makarova171.91

Yu-Na Kim of South Korea shattered the world record with a score 18 points higher than the previous record. How would the mean and median of this group change if we left out her score?

Example 3

On your first four math tests you earned a 85, 80, 95, and a 65 (hey, anyone can have a bad day). What must you earn on your next test to have a mean score of at least 80?

Exercise 1

Find the mean, median, mode and range of this data set: 

19, 18, 21, 16, 15, 17, 20, 18

Exercise 2

Which of the three averages does not describe the "middle" of this data set? 

100, 99, 97, 97, 96, 98, 95, 72

Exercise 3

On his first three quizzes, Patrick earned a 15, 18, and 16.  (A perfect score would have been 20 points.)  What does he need to earn on the next quiz to have a mean score of at least 17?