Sometimes a general description of a distribution does not give enough information, and a more precise way to talk about center or spread would be more useful. The mean, or average, is a number we can use for the center to summarize a distribution.

We can think about the mean in terms of “fair share” or “leveling out.” That is, a mean can be thought of as a number that each member of a group would have if all the data values were combined and distributed equally among the members.

For example, suppose there are 5 containers, each of which has a different amount of water: 1 liter, 4 liters, 2 liters, 3 liters, and 0 liters.

There are 5 identical tape diagrams that are each partitioned into 4 equal parts. The first diagram has 1 part shaded. The second diagram has 4 parts shaded. The third diagram has 2 parts shaded. The fourth diagram has 3 parts shaded. The fifth diagram has no parts shaded.

To find the mean, first we add up all of the values. We can think of this as putting all of the water together: $1+4+2+3+0=10$.

  

  

To find the “fair share,” we divide the 10 liters equally into the 5 containers: $10\div 5 = 2$.

The mean is useful when each unit of measurement has equal importance. For example, it may make sense to find the mean score of assignments of the same importance, such as all quizzes. If some grades are more important, it may not make sense to find the mean. For example, it may not make sense to find the mean score when there are 6 short homework assignments and one major essay.

Suppose the quiz scores of a student are 70, 90, 86, and 94. We can find the mean (or average) score by finding the sum of the scores $(70+90+86+94=340)$ and dividing the sum by four $(340 \div 4 = 85)$. We can then say that the student scored, on average, 85 points on the quizzes.

In general, to find the mean of a data set with $n$ values, we add all of the values and divide the sum by $n$.