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Lesson 3: Median in the Middle

Lesson 3: Median in the Middle

Objective

You 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.

Vocabulary

median

Essential Concepts

Lesson 2 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 for when there are outliers because it better matches what we think of as "typical".

Lesson

  1. During the previous lesson you 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.

  2. Consider the following numbers below:

    8, 2, 6, 3, 7, 4, 9, 5, 5

  3. Since there are nine numbers in the list above, you should be using the fifth number as the median. This is because it is directly in the middle, there are four numbers above it, and four numbers below it.

  4. However, you cannot simply pick the middle number of the list as it is currently written (this would give a median value of 7). Instead you must first arrange the numbers in numerical order (from lowest to highest).

    2, 3, 4, 5, 5, 6, 7, 8, 9

  5. Now you can identify that the true median value of this list of numbers is 5.

  6. Consider this scenario:

    A class of 28 students was randomly given sticky notes with the number 0 written on all of them but one, which had the number 1,000,000 on it.

  7. The class created the visualization below (a dotplot) by placing the sticky notes at the corresponding values on the x-axis. Write your answers to the following questions in your IDS Journal:

    1. What is the typical value of these data?

    2. Using the formula we learned in class, what's the mean, or average, value of this distribution?

    3. Does the mean you calculated match your understanding of “typical"?

    4. Why is the mean not capturing our notion of “typical”?

  8. Since you were introduced to the idea of the median as a measure of center at the beginning of class, what do you think will be the median value for this data set?

  9. In your IDS Journal, write the answers to the following questions:

    1. Why is there such a large difference between the mean and median values, even though they are both measures of center?

    2. Is there a specific reason why the mean is larger than the median for this particular set of data?

  10. For the next activity, click on the document name to download a fillable copy of the Medians - Dotplots or Histograms? handout (LMR_2.4).

  11. For the first two plots, write your answers to the following questions in your IDS Journal:

    1. Which plot makes it easier to find the median number of candies eaten – the dotplot or the histogram? Why?

    2. What is the median number of candies eaten?

  12. You will practice finding medians of distributions using the Where is the Middle? handout (LMR_2.5). You will be determining medians when distributions have different shapes (e.g., symmetric, left-skewed, right-skewed). Follow the instructions in your handout and complete it.

  13. Click on the document name to download a fillable copy of the Where is the Middle? handout (LMR_2.5).

  14. In your IDS Journal, answer the following questions:

    1. What did you notice about the relationship between the mean and median values for the symmetric distributions?

    2. What did you notice about the relationship between the mean and median values for the left-skewed distributions?

    3. What did you notice about the relationship between the mean and median values for the right-skewed distributions?

  15. Continue referring back to the handout that you just completed, focusing on 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?

    2. For each of the plots in the Where is the Middle? handout, which value better matches your idea of “typical” for that specific distribution?

  16. What patterns can we recognize as the better measures of center for skewed distributions? And for symmetric distributions?

  17. Consider the last 2 plots in the Medians – Dotplots or Histograms? handout. Both the dotplot and the histogram depict the number of candies eaten by a group of 330 high school students.

  18. Answer the following questions in your IDS Journal:

    1. Which plot makes it easier to find the median number of candies eaten – the dotplot or the histogram? Justify your answer.

    2. What is the median value?

Reflection

What are the essential learnings you are taking away from this lesson?

Homework

Do a quickwrite in your IDS Journal about what you think are the three most important topics you learned in this lesson.

Calculate the median values for each of the Personality Color scores.

Compare the median values to the mean values (calculated in Lesson 2). If we were to create a dotplot of the scores, what might be the shape of the distribution? Justify your answer.