Applying medians and means

1. Show slide 1 of the PowerPoint. Consider using the context of geckos as a link to learning in other curriculum areas.
   What is this creature? Have you seen a gecko like this?
   Show slide 2.
   What do you think the measurements mean?
   Agree that the measurements refer to the length of each gecko.
    
2. Ask individual students to cut a paper strip the same length as each gecko. Ask them to record the measurement on the strip. Place the strips side by side.
   What will be the average (central) length of this sample of geckos?
   How can we find this out?
   Students might suggest putting the strips in order of length and choosing the middle length. Together, do this and find the median length.
   The median length is 12cm. That’s one way to measure the centre of the lengths.
    
3. I will show you another measure of centre, called the mean. 
   Together, tape the gecko lengths end on end to form a continuous strip.
   What is the length of this strip?
   Students should realise that adding the five lengths gives a total of 14 + 10 + 9 + 12 + 15 = 60cm.
   Can anyone divide this strip into five equal lengths, by folding it into fifths?
    
4. Students might offer possible ways to form fifths:
   • Looping the strip into overlapping lengths that are equal:
     
      
   • Halving the strip and unfolding, then folding, the ends in until five equally sized pieces are created.
     
      
   • Divide the total length of 60cm by 5 to get 60 ÷ 5 = 12cm
      
5. Confirm that the folding and dividing method give the same length of 12cm.
   12cm is the mean length. We got that from adding the lengths, then dividing the total length by the number of lengths.
    
6. Show slide 3: two dot plots of the gecko lengths, one with the median showing as a vertical line. The other dot plot shows the mean.
   What do you notice about the dot plots?
   Students should notice that the median and mean are the same, and that the distribution looks balanced about the median and mean.
   Discuss why the median and mean are the same for this distribution.
   Introduce relevant te reo Māori kupu such as tau waenga (median) and toharite (mean).
    
7. Show slide 4: another dataset of gecko lengths. Make lengths for each gecko and record the length on each strip. Put the strips in order by length and find the median. There are ten geckos, so the median is equidistant between the 5th and 6th gecko (11cm).
    
8. Tape the lengths together in a continuous strip.
   What is the total length of all ten geckos? 
   Let students use a calculator, if needed, to find the total (120cm or 1.2m). Confirm the total length by measurement.
   There are ten geckos. What is the mean length? 
   What do I get if I share the 120cm equally among ten geckos?
   The division 120 ÷ 10 = 12cm gives the mean. You might confirm that by folding the whole strip into fifths, then halving the fifths to get tenths.
    
9. Show slide 5: the median and mode on the same dot plot for the second dataset of gecko lengths.
   Which line shows the mean? Which line shows the median? How do you know?
   Why are the mean and median not the same this time?
   Look for students to recognise that the distribution is non-symmetric, and the gecko with length 21 cm is an outlier.
    
10. Show slide 6: a dot plot of another dataset of gecko lengths.
    Do you think the median and mean will be equal or different?
    Calculate both measures of centre to find out.
    Which measure gives the best idea of the middle of the distribution?

Next steps

1. Use real data displayed on graphing platforms to look at median and mean as measures of centre. For example, go the random sampler at Census at School and download a sample of Year 7 and 8 students. Choose numeric variables such as height, school bag weight, screen time, and sleep time. Use a graphing tool, such as CODAP, to display the data and provide summary statistics (median and mean).

• Discuss which measure gives the best indicator of the centre of the distribution.
• Discuss why the mean and median are close, and why they may be wide apart.
   

2. Investigate situations in which mean, and median, are used in real life. Median tends to be used when the data is not symmetrically distributed, such as for house prices and income. That is because a few high house prices and incomes can make the mean an unreliable measure of centre. Mean tends to be used went the distributions are reasonably symmetric, particularly where the normal distribution applies. Mean is used to find average height, for example.