Lesson 3: Measures of Center

Property

A measure of center summarizes a data set with a single "typical" number.

• Mode: The value that occurs most often.
• Median: The exact middle value when the data is ordered from least to greatest. (If there is an even number of values, average the two middle numbers).
• Mean (Average): The sum of all values divided by the total count $n$.

When calculating the mean, you must note whether your data is a Sample (a small surveyed subset) or a Population (the entire group existing in reality).

• Population Mean is represented by the Greek letter $\mu$ (mu).
• Sample Mean is represented by $\bar{x}$ (x-bar).

Examples

• Finding the Median: For the data set {9, 2, 7, 5, 11}, order it first: {2, 5, 7, 9, 11}. The middle value is the 3rd one, so the median is 7.
• Finding the Mean: For five quiz scores {8, 10, 7, 9, 6}, the sum is 40. The mean is $40 \div 5 = 8$.
• Choosing the Best Measure: House prices on a street are 200k, 210k, 225k, 240k, and 950k. The mean is 365k, which is heavily distorted by the one 950k mansion. The median is 225k, which is a much better measure of the "typical" house on this street.

Property

The median is a number that divides an ordered data set into two parts with an equal number of values in each part.
To find the median, you must first put the values in order from lowest to highest.

• If there are an odd number of data points, the median is the number right in the middle.
• If there are an even number of data points, the median is the number halfway between the two middle values (their mean).

Examples

• For the data set {9, 2, 7, 5, 11}, we first order it: {2, 5, 7, 9, 11}. Since there are five values, the middle value is the 3rd one, so the median is 7.
• For the data set {14, 6, 8, 20}, we order it: {6, 8, 14, 20}. With an even number of values, the median is the mean of the two middle numbers: $\frac{8+14}{2} = 11$.
• The prices of five houses on a street are 200k, 210k, 225k, 240k, and 950k dollars. The median price is 225k dollars, which is a more typical value than the mean (365k dollars), which is skewed by the expensive house.

Property

The third measure of center is called the mode. This is the number that appears more often than any other number(s).

• A data set can be bimodal if two values occur with the same maximum frequency.
• If no value occurs more often than any other, there is no mode.
• The mode can be used on both numerical (quantitative) and categorical (qualitative) data.

Examples

• In the list of shoe sizes {7, 8, 9, 8, 6, 8, 10}, the number 8 appears most often. Therefore, the mode is 8.
• A class votes for their favorite pet: Dog, Cat, Fish, Dog, Cat, Bird. This data is bimodal because both Dog and Cat are the most frequent choices.
• The data set {1, 2, 3, 4, 5, 6} has no repeating values, so we say it has no mode.