Lesson 3: Median in the Middle

Objective:

Students will learn that the median is another way to measure the center, or typical-ness, of a distribution, and will understand how medians compare and contrast with the mean.

Materials:

Sticky notes (one per student)

Advanced preparation required (see step 6 below)
Poster paper
Graphics from Medians – Dotplots or Histograms? (LMR_2.4_Medians – Dotplots or Histograms)
Where is the Middle? handout (LMR_2.5_Where is the Middle)
Exported, printed, and reproduced class’s Personality Color survey data

Vocabulary:

median

Essential Concepts:

Another measure of center is the median, which can also be used to represent the typical value of a distribution. The median is preferred for skewed distributions or when there are outliers because it better matches what we think of as 'typical.'

Lesson:

Remind students that, during the previous lesson, they learned about the mean as the balancing point of a distribution and as a measure of center. In statistics, there are a few values that can be considered as measures of center – the mean is one, and another is the median. The median is the middle value in a group of ordered observations.
As a simple example, write or display the following group of numbers on the board:

8, 2, 6, 3, 7, 4, 9, 5, 5
Since there are 9 numbers in the list above, we should use the 5^th number as the median because it is directly in the middle and there are 4 numbers above it, and 4 numbers below it.
However, students should realize that they cannot simply pick the middle number of the list as it is currently written (this would give a median value of 7). Instead, they must first arrange the numbers in numerical order (from lowest to highest).

2, 3, 4, 5, 5, 6, 7, 8, 9
Now they can identify that the true median value of this list of numbers is 5.
Next, randomly distribute one sticky note to each student.

Advanced preparation required: There should be one card for every student in the class. All of the cards, except one, need to have the value 0 written on them. One card should have the value 1,000,000 written on it.
Place poster paper on the board and have the students create a dot plot by placing their sticky notes at the corresponding values on the axis. Then, ask and record answers to the following questions:
1. What is the typical value of these data? 0 – all sticky notes but one have a value of 0.
2. Using the formula we learned in class, calculate the mean, or average, value of this distribution. Answer will vary by class/class size. Example: for a class with 28 students enrolled, there would be 27 values of 0 and 1 value of 1,000,000. Therefore, the mean value would be (0*27 + 1,000,000)/28 ≈ 35,714.3.
3. Does the mean you calculated match your understanding of “typical?” Why is the mean not capturing our notion of “typical?” The 1,000,000 value is heavily skewing the calculation of the mean. It is pulling the mean to a higher value than what we consider to be typical for these data.
Since we introduced the idea of the median as a measure of center at the beginning of class, have the students find the median value of the data on their sticky notes. If time permits, have them place the sticky notes in a line across the board in order (from least to greatest) and have them find the middle number. The median value will be 0.
Ask students why there is such a large difference between the mean and median values even though they are both measures of center? Is there a specific reason why the mean is larger than the median for this particular set of data? In this case, there was an outlier value that skewed the distribution and forced the balancing point to move to the right.
Display the first 2 plots in the Medians – Dotplots or Histograms? file (LMR_2.4). They are labeled as plots for discussion for the beginning of class. Both the dotplot and histogram depict the number of candies eaten by a group of 17 high school students.

LMR_2.4
For the first 2 plots, ask students:
1. Which plot makes it easier to find the median number of candies eaten – the dot plot or the histogram? Why? The dot plot is easier because we can simply find the middle dot and record the value. It is harder on the histogram, because we would have to add up the amount in each bar to find the middle person.
2. What is the median value? The median number of candies eaten is 1 candy.
Inform the students that they will practice finding medians of distributions using the Where is the Middle? handout (LMR_2.5). They will be determining medians when distributions have different shapes (e.g., symmetric, left-skewed, right-skewed).
Distribute the Where is the Middle? handout (LMR_2.5). Students should complete the handout individually first, then compare answers with their team members. Once each team has agreed upon their answers, discuss the handout as a class.

LMR_2.5
Ask the following questions to elicit a team discussion about the relationship between means and medians:
1. What did you notice about the relationship between the mean and median values for the symmetric distributions? The mean and median values in the symmetric distributions - plots (a) and (d) - are fairly similar. For plot (a), the mean and median are exactly equal. For plot (d), the mean is actually larger than the median, but not by much (2.29 > 2).
2. What did you notice about the relationship between the mean and median values for the left-skewed distributions? The mean value was smaller than the median value in both of the left-skewed distributions - plots (c) and (f). Both plots had the same values for the mean (2.53) and the median (3.00) - clearly, the mean is much smaller than the median (2.53 < 3).
3. What did you notice about the relationship between the mean and median values for the right-skewed distributions? The mean value was larger than the median value in both of the right-skewed distributions - plots (b) and (e). For plot (b), the mean was only slightly higher than the median (1.18 > 1). For plot (e), the mean was a decent amount higher than the median (0.47 > 0).
Steer the discussion towards the relationship between the shape of a distribution and its corresponding mean and median values.
1. Is there a pattern that emerges between the mean and median values for differently shaped distributions? Yes! It seems that symmetric distributions will produce similar mean and median values, left-skewed distributions will produce smaller means and higher medians, and right-skewed distributions will produce higher means and smaller medians.
2. For each of the plots in the Where is the Middle? handout (LMR_2.5), which value better matches your idea of “typical” for that specific distribution? For plot (a), both the mean and median agree and appear to be the balancing point of the distribution – both match what we think is typical. For plot (b), the median seems to be more typical, but the values are very close. For plot (c), the median appears to be a more typical value. For plot (d), both the mean and median appear to be capturing our idea of typical. For plot (e), the median is a better match to typical. For plot (f), the median is also a better match.
Steer the discussion so that students recognize that the better measures of center for skewed distributions are typically medians, and the better measures for center for symmetric distributions are typically means.
Display the last 2 plots in the Medians – Dotplots or Histograms file (LMR_2.4). They are labeled as plots for discussion for the end of class. Both the dot plot and histogram depict the number of candies eaten by a group of 330 high school students.
For the last 2 plots, ask students:
1. Which plot makes it easier to find the median number of candies eaten – the dot plot or the histogram? Why? The histogram is easier because we can estimate based on the distribution’s shape. There are too many dots in the dot plot to find the exact middle person.
2. What is the median value? The median number of candies eaten is 7 candies.

Class Scribes:

One team of students will give a brief talk to discuss what they think the 3 most important topics of the day were.

Homework

Students should calculate the median values for each of their personality color scores. They should compare the median values to the mean values (calculated in Lesson 2) and make a decision about the possible shape of the distribution if we were to create a dot plot of the scores.