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Session 2. Reading Results: Average and Spread
Objective. Describe a set of class marks with a centre and a spread, draw it as a dot plot, and say what it means for teaching.
An average is a convenient way to say a lot with one number, and that is exactly the danger. A mean of 12 out of 20 could describe a calm class where everyone is close to 12, or a split class where half are at the bottom and half at the top. Your next lesson would be different in each case. This session gives you three habits: always pair a centre with a spread, draw the dot plot by hand, and check unusual marks before believing them.
Key ideas
Two kinds of average
- The mean is the total of all marks divided by the number of pupils. The median is the middle mark when marks are put in order.
- The mean uses every mark, so one very low or very high mark pulls it. The median ignores how far the extremes are, so it stays steady.
- When the two are far apart, look for a few unusual marks before you say anything about the class.
Spread: what the average hides
- The range is the highest mark minus the lowest. It is the simplest measure of spread and needs no calculator.
- Two classes can have the same mean and need very different teaching. Everyone near 12 is one situation; half near 6 and half near 18 is another.
- So always report two things together: a centre (mean or median) and a spread (range).
Draw it: the dot plot
- Draw a line of possible marks. For each pupil put one dot above their mark. Stack dots when marks repeat.
- In two minutes you see the shape: one hump, two humps, a long tail, or one lonely dot far from the rest.
- Two humps often mean two groups of pupils (those who understood and those who did not). One average cannot show that.
Careful reading of numbers
- Before you compute, clean: an absent pupil is not a 0. Remove them from the calculation and note how many were left out.
- Compare classes only if the same test was used under similar conditions. A mean of 12 on an easy test and 12 on a hard test are not the same.
- A number describes the class. It does not describe any one child, and it never says why the marks look as they do.
Two classes, one mean of 12
Class 8A
One tight hump
- Range 6 (9 to 15)
- One plan fits most
- Short extension for a few
Class 8B
Spread over the scale
- Range 17 (3 to 20)
- Seven at 9 or below
- Needs tiered tasks
Why the dot plot comes before the calculation
Calculation hides shape; a picture shows it. If you draw the dot plot first, you notice at once whether the marks form one hump, two humps or a long tail, and you then choose which number to compute. One hump: the mean or median describes the class well. Two humps: neither number describes anyone, and the useful fact is that there are two groups. A long tail to the left: the median is the safer centre, because a few very low marks drag the mean.
Drawing takes about two minutes for a class of 25 and needs only a ruler and a pen. Pupils can do it too: a dot plot of the class's own results is an honest, nameless way to show them where the class stands.
Unusual marks: error, circumstance or need?
A mark far from the rest has three possible explanations, and each leads to a different action. It may be a recording error (a 2 that should be 12): correct it. It may reflect a circumstance (the pupil was ill, absent, or sat the test late): leave it out of the calculation and note why. Or it may be a real need (the pupil understood almost nothing): this is the most important mark in the table, and it deserves a conversation, not a deletion.
The order matters: first check for an error, then ask about circumstances, and only then treat it as need. Many teachers jump straight to need and later find that the number was a typing slip.
Examples from the classroom
The warm-up that flattened the class
Scene. Aziz, Grade 7 in Tashkent, finds a mean of 14 out of 20 on a geometry test and feels pleased. His dot plot shows 15 pupils at 15 to 19 and five pupils at 2 to 6. The five are the same group who sat together at the back. His next lesson starts with a 10-minute catch-up at the back while the others begin a harder task.
Insight. A good mean can hide a small group in real trouble; the dot plot makes them visible.
Zero for a missing quiz
Scene. Malika, Grade 5 in Samarkand, has 9 marks of 11 to 15 and one pupil absent, recorded as 0. The mean falls from 13.0 to 11.7 and she worries that the class has slipped. The median stays at 13. She removes the 0, writes 'absent, 1 pupil' beside the table and relaxes.
Insight. The median is steady against one unusual mark, and a quick glance at it can save a false alarm.
Comparing across two tests
Scene. Dilnoza gives Grade 8 in Fergana a September test (mean 9) and a December test (mean 13) and announces big progress. A colleague asks whether the tests were equally hard. They were not: September's was longer and harder. She says 'improved on a different test' and, for a fair view, compares the dot plots and the share of pupils above 10 on common items only.
Insight. Averages from different tests are not directly comparable; compare only like with like.
Common pitfalls
- Reporting the mean alone and calling the class 'average'.
- Using the mean when one or two extreme marks pull it away from where most pupils are.
- Deleting an unusual mark without asking whether it is an error, a circumstance or a real need.
- Comparing two classes or two tests that were not equally difficult.
Two Classes, One Mean
- In pairs (10 min): in the Activity Kit are the marks (out of 20) of 20 pupils in Class 8A and 20 in Class 8B. Add each set and compute the mean (total divided by 20).
- In pairs (10 min): find the median and the range of each class.
- Alone (10 min): draw both dot plots on the grid. Mark where you see humps or lone dots.
- As a group (10 min): the next lesson covers the same topic for both classes. What would you do the same in 8A and what differently in 8B? Write two sentences. Then try the absent-pupil case in the kit.
Your facilitator will guide you through this activity in the session.
Questions to discuss
- The means are almost the same. What did the dot plots show that the means did not?
- When is the median a better 'average' than the mean?
- What teaching decision would you make for 8B that you would not make for 8A?
Reflect
- Which class of mine would look 'average' but hides two very different groups?
- Have I ever compared two classes using only the mean?
- What do I do now with a mark from a pupil who was absent?
Try it tomorrow
- Take one recent set of marks, sort them, and draw the dot plot before computing anything.
- Write 'centre X, spread Y' for that class and one thing it changes in your next lesson.
- Find the one mark furthest from the rest and check, in order: error, circumstance, need.
Further reading and links
Never report an average alone: give a centre and a spread, and draw the dot plot.