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S2 Maths | Frequency Tables and Histograms: Tutor Prep Guide | Class Boundaries, Density
Students draw a histogram like a bar chart with gaps, and assume the tallest bar means the most people. This lesson uses 40 marks and one set of unequal classes to make “area equals frequency” something they can say.
Objectives — Students can: (1) build a five-class table for 40 marks, with 4 students in 50 to 59, 9 in 60 to 69, 14 in 70 to 79, 9 in 80 to 89 and 4 in 90 to 99, state that the boundaries run from 49.5 to 99.5, and give the midpoints as 54.5 up to 94.5; (2) estimate the mean from midpoints, since 54.5 times 4 plus 64.5 times 9 plus 74.5 times 14 plus 84.5 times 9 plus 94.5 times 4 is 2,980, and 2,980 divided by 40 is 74.5, while the modal and median classes are both 70 to 79; (3) switch to frequency density when widths differ, giving 0.6, 1.5, 1.5 and 0.5 for the classes 0 to 10, 10 to 20, 20 to 40 and 40 to 60, and explain that each bar area equals its frequency.
01 Prerequisites (3-minute check)
If any of these fails, fix it before the lesson starts.
| Skill | Quick check | Pass mark |
|---|---|---|
| Mean | What is the mean of 10, 20 and 30? | Answers 20; anything else means the total or the divisor is unclear |
| Decimal multiplication | What is 54.5 times 4? | Answers 218, which is one row of the estimate |
| Division with decimals | What is 12 divided by 4, and 6 divided by 10? | Answers 3 and 0.6; the second is the entry point for frequency density |
02 Lesson flow (45 minutes)
| Time | Stage | Tutor does | Student does |
|---|---|---|---|
| 0-5 min | Data opening | Write 40 marks as one long line and ask what can be seen from it | Name two things they notice and write them down |
| 5-13 min | Classes and boundaries | Show five classes from 50 to 99 and mark boundaries at 49.5, 59.5 and so on | Write the five classes, five boundaries and five midpoints |
| 13-22 min | Table and mean | Show midpoint times frequency, sum and divide by 40 | Get 2,980 and 74.5, and circle the modal class |
| 22-31 min | Drawing the histogram | Draw five touching bars and stress that continuous data leaves no gaps | Draw one with boundaries on the x-axis, not midpoints |
| 31-40 min | Unequal widths | Use 0 to 10, 10 to 20, 20 to 40 and 40 to 60 to show frequency density | Compute 0.6, 1.5, 1.5, 0.5 and explain why the tallest bar need not hold most people |
| 40-45 min | Exit check | Collect three answers and mark correct, partly correct or wrong on the spot | Hand in three answers with one reason each |
03 Teaching script (read it out)
Scene 1: Act one: grouping is for seeing, not decorating
Forty marks in a line tell you nothing at a glance. Group them and you can say where most of the class sits. Five classes: 50 to 59, 60 to 69, 70 to 79, 80 to 89, 90 to 99. Watch the boundaries: the real limits of 50 to 59 are 49.5 and 59.5, because marks can carry decimals. The midpoint is the average of the two boundaries, so 49.5 plus 59.5 over 2 is 54.5. The five midpoints are 54.5, 64.5, 74.5, 84.5 and 94.5.
Students restate: boundaries are edges, and a midpoint is the average of two boundaries.
Scene 2: Act two: the estimated mean needs midpoints
The teacher has the raw marks but you only have the table, so each midpoint stands in for its whole class. 54.5 times 4 is 218, 64.5 times 9 is 580.5, 74.5 times 14 is 1,043, 84.5 times 9 is 760.5 and 94.5 times 4 is 378. They add to 2,980, and 2,980 over 40 is 74.5. That is an estimated mean, because we assume everyone in a class sits at the midpoint. For the median: with 40 students we need the 20th and 21st, and the cumulative frequencies run 4, 13, 27, so both fall in 70 to 79, the same class as the mode.
Students restate: the estimated mean is midpoint times frequency, and the median class comes from cumulative frequency.
Scene 3: Act three: the area of a bar is the frequency
Now suppose the widths differ: 0 to 10 holds 6, 10 to 20 holds 15, 20 to 40 holds 30 and 40 to 60 holds 10. You cannot compare heights directly, because 20 to 40 is twenty wide while the others are ten. Use frequency density, frequency divided by width: 6 over 10 is 0.6, 15 over 10 is 1.5, 30 over 20 is 1.5, and 10 over 20 is 0.5. On the chart 10 to 20 and 20 to 40 reach the same height, but look at the areas: 1.5 times 10 is 15 and 1.5 times 20 is 30, and the areas are the frequencies. So the tallest bar is not automatically the biggest group.
Students restate: frequency density times class width equals frequency, so only areas can be compared.
04 Question types and worked steps
| Type | Example | Steps | Answer |
|---|---|---|---|
| Building the table | Group 40 marks into five classes and give boundaries, midpoints and frequencies. | (1) choose classes and width (2) write boundaries by adding or subtracting 0.5 (3) tally frequencies (4) compute midpoints (5) check the total is 40 | Frequencies 4, 9, 14, 9, 4; boundaries 49.5 to 99.5; midpoints 54.5 to 94.5. |
| Estimating the mean | Estimate the mean from the table. | (1) midpoint times frequency (2) list every row (3) add (4) divide by the total frequency (5) check the answer lies within the data range | 2,980 divided by 40 is 74.5. |
| Drawing the histogram | Turn the table into a histogram and state what belongs on the x-axis. | (1) boundaries on the x-axis (2) frequency on the y-axis (3) bars touching (4) title and units | Mark 49.5, 59.5, 69.5, 79.5, 89.5, 99.5 with touching bars. |
| Frequency density | Four classes of width 10, 10, 20 and 20 hold 6, 15, 30 and 10. Find the densities. | (1) divide each frequency by its width (2) compare densities not frequencies (3) back-check with area | 0.6, 1.5, 1.5, 0.5, with areas 6, 15, 30, 10 matching the frequencies. |
Sanity check (teach this): Reasonableness check: the five frequencies must add to the total of 40, and any other total means something was missed or counted twice; the estimated mean must lie inside the data range, and here 74.5 sits between 50 and 99 inside the modal class, so 7.45 means the decimal point slipped; a boundary is half a unit beyond the class limit, so 50 to 59 runs from 49.5 to 59.5; density times width must return the frequency, and 1.5 times 20 giving 30 confirms it; continuous data has touching bars, so gaps mean a bar chart was drawn instead.
05 Independent practice (3 questions)
| Question | Answer | What to watch |
|---|---|---|
| Twenty values are grouped as 0 to 5 with 3, 5 to 10 with 7 and 10 to 20 with 10. Find the frequency densities. | 0.6, 1.4 and 1.0. | Was each frequency divided by its own width, with 10 used for the third class |
| Estimate the mean from that table. | (2.5 times 3) plus (7.5 times 7) plus (15 times 10), all over 20, giving 210 over 20 and a mean of 10.5. | Were midpoints used rather than limits, and does the total come to 210 |
| In that table, is the class with the highest frequency also the one with the highest density? | No: the highest frequency is 10 to 20 with 10, but the highest density is 5 to 10 at 1.4. | Were frequency and density kept as separate ideas |
06 Common mistakes and what to say
| Mistake | Student writes | Say this |
|---|---|---|
| Gaps in a histogram | “I drew it like a bar chart, one space between the bars.” | Say: “A histogram shows continuous data, and a mark can be 59.4 or 59.6, so there is no gap and the bars touch. Gaps mean you drew a bar chart.” |
| Limits used for the mean | “50 times 4 plus 60 times 9, then divide by 40.” | Say: “Each class needs a representative value, the midpoint 54.5. Using 50 treats every 59 as a 50 and pulls the mean down.” |
| Heights compared directly | “The 20 to 40 bar is tallest, so it holds the most people.” | Say: “20 to 40 is twenty wide while the others are ten. Divide by the width first, then multiply back: the area of 30 is the real count.” |
| Frequencies that do not total | “4 plus 9 plus 14 plus 9 plus 4 is about 40.” | Say: “It must be exactly 40. Anything else means a value was missed or double counted, so count again.” |
07 Homework (paper, section and question numbers)
- Maths index: redo practice questions 1 to 3 and draw the histogram from question 1 on squared paper — This chapter, practice 1-3
- Maths index: collect 20 values of your own, such as hours slept by people at home, then build the table, draw the histogram and estimate the mean — This chapter, section 5 frequency density question 1
08 Exit check (last 5 minutes)
| Question | Answer | Decision |
|---|---|---|
| What are the boundaries and midpoint of the class 60 to 69? | Boundaries 59.5 and 69.5, midpoint 64.5. | Answering 60 and 69 means half-unit boundaries are not understood |
| A class has frequency 12 and width 4. What is its density? | 3. | Answering 48 means multiplying instead of dividing |
| What separates a histogram from a bar chart? | A histogram shows continuous data with touching bars and boundaries on the x-axis; a bar chart shows categories with gaps. | Answering only about colours means the data type was missed |
09 Tutor FAQ
Which year is this for?
S2. S4 statistics reuses boundaries and density for standard deviation, and S6 histograms feed probability estimates.
Is 45 minutes enough for three acts?
Yes. Acts one and two take about eight minutes each, and the third is the heart of the lesson at nine; draw once only and work one density question live.
Is there practice material?
Yes. The maths index links this chapter practice with answers, and it pairs with the percentage and ratio chapter for comparing data.
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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.