Statistical data: Mean. Standard deviation

Objective:

  • To know the definitions and formulas for calculating the mean and standard deviation.

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

  • Grade 6. “Mean. Dispersion. Standard deviation.”

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:

  1. Arrange the values in ascending order.
  2. If the number of values is even, the median is the arithmetic mean of the two “middle” values.
  3. 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. The average: (170 + 175 + 180 + 185 + 190) / 5 = 180 cm.

Variation

Range is the distance between the minimum and maximum data points.

Example: Suppose we have a data set consisting of 5 measurements of people’s height: 170 cm, 175 cm, 180 cm, 185 cm, 190 cm. Range: 190-170=20.

The interquartile range (IQR) is the range from the 25th percentile to the 75th percentile, or the average of the 50th percentile of a set of numbers. It is often considered a means of determining what the average range should be.

Example: Suppose we have a data set consisting of 5 measurements of people’s height: 170 cm, 175 cm, 180 cm, 185 cm, 190 cm.

Q1=(x1+x2)/2=(175+170)/2=172.5

Q3=(x4+x5)/2=(185+190)/2=187.5

IQR=Q3-Q1=187.5-172.5=15

The mean absolute deviation is the average distance from each data point to the mean.

Example: Suppose we have a data set consisting of 5 measurements of people’s height: 170 cm, 175 cm, 180 cm, 185 cm, 190 cm. ⅕(170+175+180+185+190)=180

MAD=⅕(∣180−170∣+∣180−175∣+∣180−180∣+∣180−185∣+∣180−190∣)=⅕(10+5+0+5+10)=6.

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. Arithmetic mean and median

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

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

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

Step 3. Enable the buttons in the area displaying the data and the buttons predict median, predict mean, median, mean.

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

Step 5. Hit the ball 2-3 more times. Examine the data.

Step 6. Delete the data.

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

Part 2. Variation

Step 8. Open the “Variability” section. In the work area you are provided with:

  • Data display area: range, interquartile range, mean absolute deviation buttons;
  • Four different participants and a ball;
  • Buttons to hit the ball: hit 1 time and hit 5 times;
  • Distance: 0-15 meters;
  • Median, arithmetic mean, pointer and interval buttons;
  • Delete data and restart buttons.

Step 9. Add the median, arithmetic mean, pointer, and interval buttons.

Step 10. Hit the ball 3 times. Start the range button (range). Examine the data.

Step 11. Delete the data.

Step 12. Press the button to hit the ball 5 times. Examine the data.

Step 13. Go to the “interquartile range” chapter.

Step 14. Start the interquartile range (IQR) button. Examine the data.

Step 15. Go to the mean absolute deviation chapter.

Step 16. Run the mean absolute deviation (MAD) button. Examine the data.

Step 17. You can try the experiments several times by changing the data.

Conclusion

This virtual activity is a valuable tool for learning basic statistical concepts such as mean, median, variance, standard deviation, interquartile range and mean absolute deviation. It allows students to not only learn the definitions and formulas, but also visualize how these concepts are applied to real data. Visualizing data with graphs and charts makes learning more understandable and engaging.