Finding the Median of a Data Set

Lesson 6: The Median

• Lesson explores the median of a data set and its implications

6.1: The Plot of the Story

• Two dot plots (data set A and B) are provided with ages ranging from 10 to 75, in increments of 5
• Data set A shows 5 dots between 15-20, 7 dots between 40-45, 7 dots between 45-50, and 1 dot between 55-60
• Data set B shows 6 dots between 15-20, 2 dots between 30-35, 1 dot between 35-40, 1 dot at 40, 3 dots between 40-45, 3 dots between 45-50, and 4 dots between 60-65
• Twenty high school students, teachers, and guests attended a musical rehearsal, and the average age was 38.5 years with a MAD of 16.5 years
• Twenty people watched a high school soccer team practice one evening, with an average age of 38.5 years and a MAD of 12.7 years
• Another evening, twenty people watched the soccer team practice; the mean age was similar, but the MAD was about 20 years

6.2: Finding the Middle

• An index card is used to record the first and last name and count the number of letters in name
• Data set on numbers of siblings includes: 1, 0, 2, 1, 7, 0, 2, 0, 1, 10
• Data should be sorted from least to greatest to find the median
• Consider whether the median effectively measures typical number of siblings for group

6.3: Mean or Median?

• Six cards (dot plot or histogram) are sorted into two piles based on distributions
• Sorting decisions are discussed
• Answer questions using information on cards:
  • Card A: What is a typical age of dogs being treated at the animal clinic?
  • Card B: What is a typical number of people in Irish households?
  • Card C: What is a typical travel time for New Zealand students?
  • Card D: Is 15 years old a good description of a typical age of people at a birthday party?
  • Card E: Is 15 minutes or 24 minutes a better description of travel time to school for students in South Africa?
  • Card F: Is 21.3 years old a good description of the age of people on a field trip to Washington, D.C.?
• Consider how to decide which measure of center to use for dot plots on Cards A-C, and for Cards D-F

Additional Considerations

• Diego suggests the median is a better way to measure student performance in a course
• Teachers typically use the mean to calculate student final grades based on scores from tests, quizzes, homework, projects and assignments

Summary: Median

• The median is a measure of central distribution, the middle value in an ordered data set

• In a data set, half of the values are less than or equal to the median, and half are greater than or equal to it

• To find the median, values ordered from least to greatest, then identify the middle number

• For 5 dogs with weights of 20, 25, 32, 40, 55 pounds, the median weight is 32 pounds because three dogs weigh less than/equal to 32 pounds and three weigh greater than/equal to 32 pounds

• For 6 cats with weights of 4, 6, 7, 8, 10, 10 pounds, the median weight between 7 and 8 pounds because half the cats weigh less than/equal to 7 pounds and half weigh greater than/equal to 8 pounds

• With an even number of values, the number exactly between the two middle values is taken

• The median cat weight is 7.5 pounds in the example

• A set of 30 cookies has a mean weight of 21 grams but median weight of 23 grams

• The dot plot data for cookie weights in grams (8 through 34, in increments of 2) showed 9 grams (1 dot), 10 grams (1 dot), 13 grams (1 dot), 14 grams (1 dot), 16 grams (1 dot), 17 grams (1 dot), 19 grams (1 dot), 20 grams (2 dots), 21 grams (2 dots), 22 grams (3 dots), 23 grams (6 dots), 24 grams (5 dots), 25 grams (4 dots), 26 grams (1 dot)

• The median is closer to where data points cluster, and the median is a preferred measure of center

• The mean is influenced by values and may be farther from most data points

• In symmetrical/approximately symmetrical distributions, the mean and median are close

• If a distribution is asymmetrical, the mean and median differ; the mean is influenced by each value in the data set

• The median is reported as the typical value

Glossary Entries

• Average is another name for the mean of a data set
• The average of the data set 3, 5, 6, 8, 11, 12 is 7.5
• The mean measures a data set's center and can be thought of as a balance point
• The mean of 7, 9, 12, 13, 14 is 11
• The mean is calculated by adding all numbers in data set, then dividing by how many numbers there are
• Mean Absolute Deviation (MAD) measures how spread out a data set is
• For the data set 7, 9, 12, 13, 14, the MAD is 2.4, meaning travel times are typically 2.4 minutes away from the mean (11)
• The MAD is calculated by adding the distance between each data point and the mean, dividing by how many numbers there are
• A measure of center is a value seems typical for a data distribution; mean and median are measures of center
• The median measures the center of a data set; it is the middle number when values are listed in order
• For the data set 7, 9, 12, 13, 14, the median is 12
• For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle, and the median is the average of those two numbers