2.6 Measures of Center
Let’s keep working through the acronym SOCS for describing key aspects of our data, this time focusing on the center.
- Shape
- Outliers
- Center
- Spread
The “center” is a way of describing the “central tendency” or “typical value” of a dataset. The two most widely used measures of the center of the data are the mean (average) and the median. Most people are familiar with the ideas of these two: (1) to calculate the mean weight of 50 people, add the 50 weights together and divide by 50, and (2) to find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts.
However, some datasets may be better summarized by one or the other. The most “appropriate” measure of center depends on the shape of the distribution and the presence of extreme values or potential outliers.
The Mean
The mean is the most common measure of the center. The words “mean” and “average” are often used interchangeably. The technical term is “arithmetic mean,” and “average” technically refers to a center location. However, in practice among non-statisticians, “average” is commonly accepted for “arithmetic mean.”
When each value in the dataset is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The sample mean is denoted by an x with a bar over it, , pronounced simply “x bar.”
The Greek letter μ (pronounced “mew”) represents the population mean. We will often use the sample mean to estimate the population mean. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample to be taken truly at random.
Example
Calculate the mean of the sample: 1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4.
Solution
= (1+1+1+2+2+3+4+4+4+4+4)/11 = 2.7
Your Turn!
Calculate the mean of the sample: 7, 10, 14, 14, 15, 21, 38, 38, 38, 56.
Solution
= (7+10+14+14+15+21+38+38+38+56)/10 = 25.1
The Median
The median is generally a better measure of the center when there are extreme values or outliers because it is more robust, or not affected by the precise numerical values of those outliers.
Especially for larger datasets, you may choose to use the following location function over the traditional counting method to find the median:
.
Remember that this function simply tells you where to look for the median, not the actual value itself, and n is the total number of data values in the sample (sample size).
Once you a have arranged your data in ascending order (smallest to largest), the method of finding your median will differ slightly based on whether you have an odd or even sample size. If n is odd, the median is included in the dataset and is simply the middle value found using the location function. If n is an even number, your location function will give you a decimal value ending in .5, and to find the median, you must calculate the average of the numbers in the and + 1 positions.
For example, if