The "center" of a data set is also a way of describing location. The two most widely used measures of the "center" of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts (previously discussed under box plots in this chapter). The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.

The mean can also be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an xx with a bar over it (pronounced "xx bar"): x¯ x .

The Greek letter μμ (pronounced "mew") represents the population mean. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random.

To see that both ways of calculating the mean are the same, consider the sample:

1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4

In the second example, the frequencies are 3, 2, 1, and 5.

You can quickly find the location of the median by using the expression n + 1 2 n + 1 2 .

The letter nn is the total number of data values in the sample. If nn is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If nn is an even number, the median is equal to the two middle values added together and divided by 2 after the data has been ordered. For example, if the total number of data values is 97, then n + 1 2 n + 1 2 = 97 + 1 2 97 + 1 2 = 4949. The median is the 49th value in the ordered data. If the total number of data values is 100, then n + 1 2 n + 1 2 = 100 + 1 2 100 + 1 2 = 50.550.5. The median occurs midway between the 50th and 51st values. The location of the median and the value of the median are not the same. The upper case letter MM is often used to represent the median. The next example illustrates the location of the median and the value of the median.

# Enter the AIDS data
aids = c(3, 4, 8, 8, 10, 11, 12, 13, 14, 15, 15, 16,
    16, 17, 17, 18, 21, 22, 22, 24, 24, 25, 26, 26,
    27, 27, 29, 29, 31, 32, 33, 33, 34, 34, 35, 37,
    40, 44, 44, 47)

# mean and median
mean(aids)
## [1] 23.57
median(aids)
## [1] 24

# summary statistics
summary(aids)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
##     3.0    15.0    24.0    23.6    32.2    47.0

Another measure of the center is the mode. The mode is the most frequent value. If a data set has two values that occur the same number of times, then the set is bimodal.

# Enter the exam scores data
exam.scores = c(50, 53, 59, 59, 63, 63, 72, 72, 72, 
    72, 72, 76, 78, 81, 83, 84, 84, 84, 90, 93)

# Use 'table' to find the mode
table(exam.scores)
## exam.scores
## 50 53 59 63 72 76 78 81 83 84 90 93 
##  1  1  2  2  5  1  1  1  1  3  1  1

# That output is difficult to scan. Sort the
# output to make it easier to identify the most
# frequently occurring value.
sort(table(exam.scores))
## exam.scores
## 50 53 76 78 81 83 90 93 59 63 84 72 
##  1  1  1  1  1  1  1  1  2  2  3  5
# 72 occurs 5 times, making it the mode.

The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean x ¯ x ¯ of the sample is very likely to get closer and closer to µµ. This is discussed in more detail in The Central Limit Theorem.

You can think of a sampling distribution as a relative frequency distribution with a great many samples. (See Sampling and Data for a review of relative frequency). Suppose thirty randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below.

If you let the number of samples get very large (say, 300 million or more), the relative frequency table becomes a relative frequency distribution.

A statistic is a number calculated from a sample. Statistic examples include the mean, the median and the mode as well as others. The sample mean x ¯ x ¯ is an example of a statistic which estimates the population mean μμ.