Applying measures of centre and spread to distributions

1. Show the students Slide 1 of the PowerPoint. The slide shows two dot plots of sleep time for a sample of 100 students. The data is sorted into groups by year level.
   What do these dot plots show?
   Students should discuss the variables of interest and sleep time.
   How is sleep time measured? (in hours)
   They should notice that there is a separate dot plot for each of years 7 and year 8.
    
2. Ask: Are there similarities and differences between the sleep times of males and females?
   Let students discuss, in pairs, the question and share what they notice. Pair students to encourage tuakana-teina and productive learning conversations. Points might include:
   • The shapes of the distributions are similar. Both have a single mode (most common time) and are approximately symmetric.
   • The sleep times for males and females have similar spread between lowest and highest number of hours.
   • There are more females than males in the sample.
   • The centres for males and females are hard to find visually though they appear to be quite similar.
   • You might introduce relevant te reo Māori kupu such as tau waenga (median), tau tānui (mode) and toharite (mean).
      
3. Ask students to predict the central hours of sleep visually (about 9 hours).
   Slides 2 and 3 show the median and mean for the distributions.
   Here are the medians. How did the computer find the median of each distribution?
   Students should know that the middle point was identified by ordering the times.
   What do you notice?
   Students should comment that the median is higher for year 7s by about half an hour.
    
4. Can we say that students in Year 7 get more sleep than students in year 8?
   Students should raise points like:
   • At an individual level you cannot say that a year 7 gets more sleep than a year 8 since there is considerable overlap of the two distributions.
   • You can say that the average amount of sleep for males is slightly higher than the average amount for females.
      
5. Slide 3 shows the means.
   Do the means match what we saw with the medians? Why are the measures very close?
   Since the distributions are both symmetric the median and mean are expected to be close. Again the average sleep time for males is slightly higher than the average for females.
    
6. Slides 4 -7 show two further examples of comparing the distributions of different groups. Use the first slide in each pair to ask students what they notice in the dotplots.
   Use the second slide in the pair to show the median and mean for each distribution. Look for students to:
   • predict whether the median and mean will be close or apart.
   • make comparative statements in context.
   • justify the statements using reference to the statistical features (mean, median and range).

Next steps

1. Provide open problems for students to solve that involve median and mode.
   • Create a dataset with a median of 15.
   • Create a dataset with a mean of 100.
   • Create a dataset with a median on 30 and a mean of 20.
      
2. Explore the use of digital technology to look for patterns and differences between groups. Begin a Statistical Inquiry Cycle by posing questions about topics of interest to your students. Ideally questions can be answered by gathering or accessing numeric data.
   Examples might be:

• Do Year 10 students get more pocket money than Year 8 students?
• Do students outside of Auckland have faster reaction times than students in Auckland
• Can females shoot better at netball than males?