In this section, you will determine and interpret the center of numerical data that are represented in stem-and-leaf plots and dot plots. When you interpret the center of data, you are referring to either the mean or the median of the data set.

Practice finding the mean using Mr. Smith’s fourth period ELA class’ 10 question assessment scores.

The students’ assessment scores are as follows.

70, 60, 80, 90, 50, 100, 80, 90, 80, 70, 50, 90, 100, 90, 80, 70, 70, 80, 90, 80, 100, 100, 70, 80, 80

Follow the steps below to find the mean of the scores. Click on the Continue button to begin the animation.

Use the information from the interactive to answer the following questions.

• Is the mean reasonable? Explain.

  Interactive popup. Assistance may be required.

  Need a hint?

  The mean is also the average, a value that would be near the middle of most of the numbers, a general representative of the number values.
  Interactive popup. Assistance may be required.

  Check Your Answer

  Yes, it is reasonable. There are several 80’s and a significant number of values close to 80.
• Describe how knowing the mean will help Mr. Smith know if the students understood the material.

  Interactive popup. Assistance may be required.

  Need a hint?

  If the mean is the center, the average or a general representation of the classes’ understanding of the material, did the students understand the material?
  Interactive popup. Assistance may be required.

  Check Your Answer

  If Mr. Smith knows that the mean of the class assessment scores is 80, he knows the average score for the class is 80. It doesn’t give him an accurate measure for the class scores, but it tells him that the average of the class is passing.

Success in providing students with a good education is when each demographic area of the student body is successful. Below are the national graduation rates for African Americans from 2006 to 2012. Follow the steps below to create a stem-and-leaf plot for the information.

Source: Pros and Cons of the U.S. Higher Education System

Use the stem and leaf plot to answer the following.

• Find the mean graduation rate.

  Interactive popup. Assistance may be required.

  Need a hint?

  In order to find the mean, add the percentages together and then divide by 7, the number of percentages provided.
  Interactive popup. Assistance may be required.

  Check Your Answer

  59 59 61 64 66 67 68 = 444
                      444 ÷ 7 = 63.4%

Click below to find the median for the information above.

• The mean and the median are used to discuss the center of the data. The median is the middle value of the number set of the sorted data. Explain why it is easier to sort the data before finding the median.

  Interactive popup. Assistance may be required.

  Need a hint?

  The median is the number in the middle, how would you find the number in the middle?
  Interactive popup. Assistance may be required.

  Check Your Answer

  One way to find the number in the middle is to make sure the data is in order from smallest to largest and then eliminate a term from both ends because it is easier to know which values to eliminate without eliminating the wrong values.
• Suppose in the above example there were two values left instead of one. How would you find the median?

  Interactive popup. Assistance may be required.

  Need a hint?

  The median is considered part of the center or middle of the data. How would you find the middle when two values are given?
  Interactive popup. Assistance may be required.

  Check Your Answer

  Add the two values together and divide by two.

Practice

A president that is elected to a four-year term has to be at least 35 years old. Below is a dot plot showing the ages in years of the presidents on the day of their inauguration.

1. Find the mean age of a US president on the day of his inauguration.

   Interactive popup. Assistance may be required.

   Need a hint?

   In order to find the mean, add the ages together and then divide by the number of presidents we have had, 44.
   Interactive popup. Assistance may be required.

   Check Your Answer

   42 + 43 + 2(46) + 2(47) + 48 + 2(49) + 50 + 5(51) + 2(52) + 5(54) + 4(55) + 3(56) + 4(57) + 58 + 60 + 3(61) + 62 + 2(64) + 2(68) + 69 =
   
   42 + 43 + 92 + 94 + 48 + 98 + 50 + 255 + 104 + 270 + 220 + 168 + 228 + 58 + 60 + 183 + 62 + 128 + 69 = 2408
                                                                                                                                                                2408 ÷ 44 = 54.72
2. Look at the data. What would be a reasonable estimate for the median? Explain.

   Interactive popup. Assistance may be required.

   Need a hint?

   Since the ages are already sorted in order, look at the spread of the data points. There are a few points at each end, but notice that most of the data points are at the center of the data.
   Interactive popup. Assistance may be required.

   Check Your Answer

   A general reasonable estimate of the center of the data would be between 51 and 57, but a closer estimate would be between 54 and 56.
3. Find the median age of a U.S. President on the day of his inauguration.

   Interactive popup. Assistance may be required.

   Need a hint?

   The numbers are sorted in order. Find the number that is in the middle of the group of data. Looking at the dot plot, a reasonable estimate would be 51 and 57, which is where most of the numbers are located.
   Interactive popup. Assistance may be required.

   Check Your Answer

   Start crossing out the lowest and highest age. The two that remain are 54 and 55. To find the median, add the two numbers together and divide by 2.
   54 + 55 = 109
   109 ÷ 2 = 54.5
   The median age is 54.5.