Section 13.4 Measures of Central Tendency

What You Will Learn Averages Mean Median Mode Midrange Quartiles

Measures of Central Tendency An average is a number that is representative of a group of data. There are at least four different averages: the mean, the median, the mode, and the midrange. Each is calculated differently and may yield different results for the same set of data.

Measures of Central Tendency Each will result in a number near the center of the data; for this reason, averages are commonly referred to as measures of central tendency.

Mean (or Arithmetic Mean) The mean, , is the sum of the data divided by the number of pieces of data. The formula for calculating the mean is where Σx represents the sum of all the data and n represents the number of pieces of data.

Example 1: Determine the Mean Determine the mean age of a group of patients at a doctor’s office if the ages of the individuals are 28, 19, 49, 35, and 49. Solution

Median The median is the value in the middle of a set of ranked data.

Example 2: Determine the Median Determine the median age of a group of patients at a doctor’s office if the ages of the individuals are 28, 19, 49, 35, and 49. Solution Rank the data from smallest to largest. 19 28 35 49 49 35 is in the middle, 35 is the median.

Example 3: Determine the Median of an Even Number of Pieces of Data Determine the median of the following sets of data. a) 9, 14, 16, 17, 11, 16, 11, 12 b) 7, 8, 8, 8, 9, 10

Example 3: Determine the Median of an Even Number of Pieces of Data Solution 9, 11, 11, 12, 14, 16, 16, 17 8 pieces of data Median is half way between middle two data points 12 and 14 (12 + 14)÷2 = 26 ÷ 2 = 13

Example 3: Determine the Median of an Even Number of Pieces of Data Solution 7, 8, 8, 8, 9, 10 6 pieces of data Median is half way between middle two data points 8 and 8 (8 + 8)÷2 = 16 ÷ 2 = 8

Mode The mode is the piece of data that occurs most frequently.

Example 4: Determine the Mode Determine the mean age of a group of patients at a doctor’s office if the ages of the individuals are 28, 19, 49, 35, and 49. Solution The age 49 is the mode because it occurs twice and the other values occur only once.

Midrange The midrange is the value halfway between the lowest (L) and highest (H) values in a set of data.

Example 5: Determine the Midrange Determine the midrange age of a group of patients at a doctor’s office if the ages of the individuals are 28, 19, 49, 35, and 49. Solution

Measures of Position Measures of position are often used to make comparisons. Two measures of position are percentiles and quartiles.

Percentiles There are 99 percentiles dividing a set of data into 100 equal parts.

Percentiles A score in the nth percentile means that you out-performed about n% of the population who took the test and that (100 – n)% of the people taking the test performed better than you did.

Quartiles Quartiles divide data into four equal parts: The first quartile is the value that is higher than about 1/4, or 25%, of the population. It is the same as the 25th percentile.

Quartiles The second quartile is the value that is higher than about 1/2 the population and is the same as the 50th percentile, or the median. The third quartile is the value that is higher than about 3/4 of the population and is the same as the 75th percentile.

Quartiles

To Determine the Quartiles of a Set of Data 1. Order the data from smallest to largest.

To Determine the Quartiles of a Set of Data 2. Find the median, or 2nd quartile, of the set of data. If there are an odd number of pieces of data, the median is the middle value. If there are an even number of pieces of data, the median will be halfway between the two middle pieces of data.

To Determine the Quartiles of a Set of Data 3. The first quartile, Q1, is the median of the lower half of the data; that is, Q1, is the median of the data less than Q2.

To Determine the Quartiles of a Set of Data 4. The third quartile, Q3, is the median of the upper half of the data; that is, Q3 is the median of the data greater than Q2.

Example 8: Finding Quartiles Electronics World is concerned about the high turnover of its sales staff. A survey was done to determine how long (in months) the sales staff had been in their current positions. The responses of 27 sales staff follow. Determine Q1, Q2, and Q3.

Example 8: Finding Quartiles 25 3 7 15 31 36 17 21 2 11 42 16 23 16 21 9 20 5 8 12 27 14 39 24 18 6 10 Solution List data from smallest to largest. 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 21 21 23 24 25 27 31 36 39 42

Example 8: Finding Quartiles Solution 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 21 21 23 24 25 27 31 36 39 42 The median, or middle of the 27 data points is Q2 = 17. The median, or middle of the lower 13 pieces of data is Q1 = 9. The median, or middle of the upper 13 pieces of data is Q3 = 24.