Measures of location: Mean Unlike the median, the mean uses all the data. This quantity is usually referred to as an average. The mean is that value which all the data in a set would be, if they were made to be equal through redistribution. A mean is calculated as follows: 𝑥 = 𝑥 1 + 𝑥 2 + 𝑥 3 +…+ 𝑥 𝑛 𝑛 = 𝑥 𝑖 𝑛 𝑥 is read as “x bar”.

The following result is important, because the sum is always equal to 0, that is: 𝑖=1 𝑛 𝑥 𝑖 − 𝑥 =0 Proof: 𝑖=1 𝑛 𝑥 𝑖 − 𝑥 = 𝑥 1 − 𝑥 + 𝑥 2 − 𝑥 +…+ 𝑥 𝑛 − 𝑥 𝑖=1 𝑛 𝑥 𝑖 − 𝑥 = 𝑥 1 + 𝑥 2 +…+ 𝑥 𝑛 −𝑛 𝑥 But 𝑥 = 𝑥 1 + 𝑥 2 +…+ 𝑥 𝑛 𝑛 → 𝑛 𝑥 = 𝑥 1 + 𝑥 2 +…+ 𝑥 𝑛 as expected.

𝑥 = 𝑥 1 𝑓 1 + 𝑥 2 𝑓 2 + … + 𝑥 𝑛 𝑓 𝑛 𝑓 1 + 𝑓 2 + … + 𝑓 𝑛 Calculating the mean from a frequency table: The mean 𝑥 , of a data set in which the variable takes the value 𝑥 1 with frequency 𝑓 1 , value 𝑥 2 with frequency 𝑓 2 , and so on, is given by: 𝑥 = 𝑥 1 𝑓 1 + 𝑥 2 𝑓 2 + … + 𝑥 𝑛 𝑓 𝑛 𝑓 1 + 𝑓 2 + … + 𝑓 𝑛 If the data in a frequency table are grouped, we need a single value to represent each class. This is called the mid-class value. The mid-class value is the mean of the class boundaries of each given class. For example, if a class range is 54.5 −59.5, the mid-class value is 57: 54.5+59.5 2 =57

Making the calculation of the mean easier. If you are given the set 907, 908, 898, 902, 897 you can find the mean by adding all the numbers and dividing by 5, or you could subtract 900 from all the numbers to get: 7, 8, −2, 2,−3 Now the mean of this set is 12 5 =2.4. Adding back 900, you have the mean of the original set: 902.4. The general formula is: 𝑥 = 𝑥−𝑘 𝑛 +𝑘, where 𝑘 is a constant.

Now do Exercise 2B. 