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State Board · Class 8 · Mathematics

Lesson 11: Statistics

3 concepts · 24 practice questions with answers · Free to read

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Frequency table and mean

From daily life

A teacher noted game scores of 20 students: 3, 5, 7, 3, 2 students scored 1, 2, 3, 4, 5 respectively. Instead of adding every score one by one, make a table — the work becomes easy.

Simple definition

The number of times a value occurs is its frequency (f). The total of all frequencies is N. The mean (average) is the sum of all values ÷ the number of values.

31527334252.8

In the figure: Marks 1 to 5 with frequencies 3, 5, 7, 3, 2 (bar heights): N = 20, Σ x × f = 56 ⇒ mean = 56 ÷ 20 = 2.8.

Formula: Mean = (sum of value × frequency) ÷ N

In words: Steps: 1) Make the table. 2) In each row work out value × f. 3) Add them all. 4) Divide by N.

Table

Score (x)Frequency (f)x × f
133
2510
3721
4312
5210
TotalN = 2056
  • Easy: Mean = 56 ÷ 20 = 2.8 marks.
  • Medium: Values 10, 20, 30 with frequencies 2, 5, 3. x × f: 20, 100, 90 ; sum 210 ; N = 10 ; mean = 210 ÷ 10 = 21.
  • Exam-type: The mean of 5 numbers is 12. A sixth number 18 is added. Old sum 5 × 12 = 60 ; new sum 60 + 18 = 78 ; new mean = 78 ÷ 6 = 13.
Live figure

Frequency table and mean

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Remember

Mean = sum of all values ÷ number of values; with frequencies it is the sum of (x × f) ÷ N.

Common mistake

The sum of frequencies is N; the sum of x values is not N.

Shortcut

The mean always lies between the smallest and the biggest value. If not, there is a mistake. Memory clue: Mean = sum ÷ count: "sum divided by count".

Spot the difference

PointSimple meanMean with frequencies
Wheneach value occurs oncevalues repeat
Formulasum ÷ countΣ(x × f) ÷ N
Example4, 5, 6, 5 → 51, 2, 3 with f = 2, 3, 5 → 2.3
Divisornumber of valuesN = sum of f

Practice questions

Easy
  1. 1Mean of 4, 5, 6, 5?
  2. 2x: 1, 2, 3 and f: 2, 3, 5. Mean?
  3. 3The mean of 6 numbers is 10. Their sum?
Medium
  1. 4x: 10, 20, 30, 40 and f: 1, 2, 3, 4. Mean?
  2. 5The mean of 5 numbers is 8. If 20 is added, the new mean?
  3. 6The mean of 7 numbers is 14. If the number 20 is removed, the new mean?
Challenging
  1. 7x: 5, 6, 7, 8 and f: 4, 6, 6, 4. Mean?
  2. 8The average of 4 tests is 72. How many marks in the 5th test make the average 75?
Show answers (try them yourself first)
  1. 1(4 + 5 + 6 + 5) ÷ 4 = 20 ÷ 4 = 5.
  2. 2x × f: 2, 6, 15; sum 23; N = 10; mean = 2.3.
  3. 36 × 10 = 60.
  4. 4x × f: 10, 40, 90, 160; sum 300; N = 10; mean = 30.
  5. 5Old sum 40; new sum 60; new mean = 60 ÷ 6 = 10.
  6. 6Sum 98; 98 − 20 = 78; new mean = 78 ÷ 6 = 13.
  7. 7x × f: 20, 36, 42, 32; sum 130; N = 20; mean = 6.5.
  8. 85 × 75 = 375; 4 × 72 = 288; 375 − 288 = 87 marks.

Subdivided bar graph

From daily life

Boys and girls in classes 5 to 8. If we draw one bar for each class and show boys and girls in different colours inside it, both the total and the shares are seen at a glance.

Simple definition

A graph in which each bar is split into two or more coloured parts is a subdivided bar graph. The full height of a bar is the total.

222446730205083525609

In the figure: Subdivided bars: lower part (blue) + upper part (orange) = total (the bar height): 22 + 24 = 46.

In words: Steps: 1) Choose a scale. 2) Draw a bar as tall as the total. 3) Show the first part, then the second part, in different colours. 4) Add a colour key.

Live figure

Subdivided bar graph

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  • Easy: Class 7: 22 boys and 24 girls. Total = 22 + 24 = 46. The bar is 46 units tall: 22 for boys at the bottom, 24 for girls on top.
  • Medium: 2022: A = 30, B = 20, total 50. 2023: A = 35, B = 25, total 60. The 2023 bar is 60 − 50 = 10 units taller.
  • Exam-type: Three parts P = 12, Q = 18, R = 10 in one bar: total = 12 + 18 + 10 = 40; from the bottom P (0 to 12), Q (12 to 30), R (30 to 40).
Remember

Draw all bars on the same scale so that comparison is fair.

Common mistake

The bar height is the total, not just the top part.

Shortcut

Height of the top part = total − sum of the lower parts. Memory clue: In a sub-divided graph the parts are stacked one over the other; the height grows by addition.

Spot the difference

PointSimple bar graphSubdivided bar graph
Each bar hasone parttwo or more parts
Heightnumber in one grouptotal number
Coloursonea different one for each part
Exampleboys 22boys 22 + girls 24 = 46

Practice questions

Easy
  1. 1Boys 18 and girls 22. Bar height?
  2. 2A bar of total 50; the first part is 20. The second part?
  3. 32021: A = 25, B = 15; 2022: A = 30, B = 20. How much taller is the 2022 bar?
Medium
  1. 4Three parts 12, 18, 10 in one bar. Total height?
  2. 5Total 60 and A = 35. B?
  3. 62020: boys 30, girls 20; 2021: boys 36, girls 24. The boys-girls difference each year?
Challenging
  1. 7A bar of total 50 is drawn at the scale 1 cm = 10 units. Its height in cm?
  2. 8Year 1: A = 20, B = 30; year 2: A = 30, B = 45. In which year is the share of B larger?
Show answers (try them yourself first)
  1. 118 + 22 = 40.
  2. 250 − 20 = 30.
  3. 32021 total 40; 2022 total 50; difference 10.
  4. 412 + 18 + 10 = 40.
  5. 560 − 35 = 25.
  6. 62020: 30 − 20 = 10; 2021: 36 − 24 = 12.
  7. 750 ÷ 10 = 5 cm.
  8. 8Year 1: 30 ÷ 50 × 100 = 60 %; year 2: 45 ÷ 75 × 100 = 60 %; the same in both years.

Percentage bar graph

From daily life

One class has 18 boys and 12 girls, another 45 boys and 30 girls. The numbers differ but boys are 60 % in both. A percentage bar graph shows this immediately.

Simple definition

A subdivided bar graph drawn with each part converted into a percent of the total. All bars are 100 % tall.

60 %40 %18 + 1260 %40 %45 + 30100

In the figure: Percentage bars: every bar is 100 %; 18 + 12 → 60 % and 40 %, 45 + 30 → the same 60 % and 40 %.

Formula: Percent = part ÷ total × 100

Live figure

Percentage bar graph

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  • Easy: 40 cups of tea and 10 of coffee, total 50. Tea = 40 ÷ 50 × 100 = 80 % ; coffee = 10 ÷ 50 × 100 = 20 %.
  • Medium: Class A: 18 boys, 12 girls; total 30; boys = 18 ÷ 30 × 100 = 60 %; girls = 40 %.
  • Exam-type: Class B: 45 boys, 30 girls; total 75; boys = 45 ÷ 75 × 100 = 60 %; girls = 40 % — different numbers but the same percentages.
Remember

In a percentage bar graph every bar is 100 % tall.

Common mistake

In a percentage bar graph all bars have the same height, so you cannot read the totals from it.

Shortcut

Check that the percentages of all parts add up to 100 %. Memory clue: In a percentage graph all bars are equally tall, because each totals 100 %.

Spot the difference

PointSimple barSubdividedPercentage
Bar hasone parttwo or more partstwo or more parts
Bar heightthe numberthe total100 % for all
Used forcomparing numberstotal and partscomparing shares

Practice questions

Easy
  1. 1Make a frequency table for 2, 3, 3, 4, 3, 2, 3, 4 and find the mean.
  2. 215 out of 60 students are absent. What percent?
  3. 340 cups of tea, 10 of coffee. Percent of tea?
Medium
  1. 4Find the mean for x: 10, 20, 30 with f: 2, 5, 3.
  2. 5A class has 18 boys and 12 girls. Percent of each?
  3. 62022: A 30, B 20; 2023: A 35, B 25. Which year has the taller total bar and by how much?
Challenging
  1. 7x: 1, 2, 3, 4 with f: 4, 6, 8, 2. Mean?
  2. 8The mean of 5 numbers is 12. After adding 18, what is the new mean?
Show answers (try them yourself first)
  1. 12 → 2 times, 3 → 4 times, 4 → 2 times; N = 8 ; sum = 4 + 12 + 8 = 24 ; mean = 24 ÷ 8 = 3.
  2. 215 ÷ 60 × 100 = 25 %.
  3. 380 %.
  4. 4210 ÷ 10 = 21.
  5. 5Boys 18 ÷ 30 × 100 = 60 % ; girls 12 ÷ 30 × 100 = 40 %.
  6. 62023, by 60 − 50 = 10.
  7. 7N = 20 ; x × f = 4 + 12 + 24 + 8 = 48 ; mean = 48 ÷ 20 = 2.4.
  8. 8(60 + 18) ÷ 6 = 13.