Statistical data: arithmetic mean, median

Objective:

  • To know the definitions of the arithmetic mean of several numbers, median; 
  • Calculate statistical quantitative characteristics.

This virtual activity is designed for use in math lessons on the following topics:

  • Grade 6. “Statistical data and their characteristics: arithmetic mean, mode, median, Kulash”

Theoretical part

Median

The median is the number that is in the middle of an ascending-ordered list of numbers. If the number of numbers is even, the median is the arithmetic mean of the two center numbers.

How to find the median:

  • Arrange the values in ascending order.
  • If the number of values is even, the median is the arithmetic mean of the two “middle” values.
  • If the number of values is odd, the median is the value that is in the middle.

Example: Suppose we have a dataset of 5 height measurements of people: 170 cm, 175 cm, 180 cm, 185 cm, 190 cm. In a data set of 5 measurements of people’s height, the median is 180 cm.

Arithmetic mean

The arithmetic mean is a number that shows the average of all the numbers in a group. To find the arithmetic mean, you need to add up all the numbers and divide the resulting sum by the number of those numbers.

Formula:

Average = (Sum of all values) / (Number of values)

Example: Suppose we have a data set of 5 measurements of people’s height: 170 cm, 175 cm, 180 cm, 185 cm, 190 cm. Average: (170 + 175 + 180 + 185 + 185 + 190) / 5 = 180 cm.

Virtual Experiment

This simulation allows students to explore the mean, the median, by working with small data sets with points far apart. Using the median screen, students can see how a data point affects the value of the median.

In the mean and median screen, students compare the mean and median. They can see how they are affected by new points or the movement of individual points.

Progression:

Section 1: Median

Step 1. Start the simulation: you will be given 3 different modes: ‘Median’, ‘Mean&Median’ and ‘Variability’. In this work, you will be working in the first two sections. Open the “Median” section.

Step 2. In the work area you are provided with:

  • Data display area: distance, data sorting, median buttons;
  • The contestant and the ball;
  • Ball kick buttons: kick 1 time and kick 5 times;
  • Distance: 0-15 meters;
  • Predict median and median buttons;
  • Data delete and restart buttons.

Step 3. Add data sort, median, predict median and median buttons.

Step 4. Press the hit the ball button 1 time. Examine the data. 

Step 5. Hit the ball 2 more times. Examine the data.

Step 6. Delete the data.

Step 7. Press the button to hit the ball 5 times. Examine the data. 

Section 2: Arithmetic Mean & Median

Step 8. Open the Mean&Median section. In the work area you are provided with:

  • Data display area: distance, arithmetic mean, median buttons;
  • Participant and ball; 
  • Ball kick buttons: kick 1 time and kick 5 times;
  • Distance: 0-15 meters;
  • Predict median, predict mean, median and mean buttons;
  • Data delete and restart buttons.

Step 9. Enable the buttons in the area displaying the data and the predict median, predict mean, median, mean buttons.

Step 10. Press the hit the ball button 1 time. Examine the data. 

Step 11. Hit the ball 2-3 more times. Examine the data.

Step 12. Delete the data.

Step 13. Press the button to hit the ball 5 times. Examine the data.

Conclusion

Two ways of evaluating data were explored in the simulation: the median and the arithmetic mean.

Comparison of the median and the mean:

  • The median is easier to understand and interpret.
  • The median is more resistant to abnormal values.
  • The arithmetic mean is more sensitive to changes in the data.
  • The arithmetic mean can be a more accurate estimate of the central tendency of the data if it is normally distributed.