Data Handling: Graphs, Averages and Probability

What to remember

• Data is shown by tally marks, pictographs, bar graphs, double bar graphs, histograms and pie charts. Each has a rule: a key for a pictograph, equal gaps for a bar graph, no gaps for a histogram, and 360° for a pie chart.
• Mean = sum of observations ÷ number of observations. Median is the middle value of ordered data. Mode is the value that occurs most often.
• Probability of an event = number of favourable outcomes ÷ total number of outcomes. It lies from 0 to 1.

Collecting and organising data

Data is a collection of facts or numbers. Data collected first-hand and not yet arranged is raw data. We arrange it in a frequency distribution table. Frequency is the number of times a value occurs.

Tally marks: We count in groups of five. Four vertical strokes are crossed by a fifth slanting stroke (shown as 卌 here), so one group stands for 5. Example: 12 is written as two groups of five and two more strokes.

The total frequency is 12 + 6 + 16 = 34.

Range = highest value − lowest value. For 12, 7, 15, 9 and 21 the range is 21 − 7 = 14.

Pictograph

A pictograph uses pictures or symbols to show data. Always read the key first. If one picture stands for 10 books, then 3½ pictures show 35 books. A half picture shows half the value. Pictographs suit Classes III and IV, where children read data easily without numbers.

Bar graph

A bar graph shows data with bars of equal width. Bars may be vertical or horizontal. They have equal gaps between them. The height (or length) of a bar shows the value. The scale must be the same all along the axis. Choose a scale that fits the data: if the largest value is 80, a scale of 1 cm = 10 units is suitable.

Double bar graph: It shows two sets of data side by side for comparison, for example the marks of boys and girls in each subject. Use two different colours or patterns and a legend. Read the pairs of bars together.

Histogram

A histogram shows grouped data (class intervals) such as marks 0–10, 10–20, 20–30. Bars touch each other because the classes are continuous. The width of a bar is the class size and the height is the frequency. This is the difference from a bar graph:

Class mark = (upper limit + lower limit) ÷ 2. For the class 10–20 it is 15. Class size = upper limit − lower limit.

Pie chart (circle graph)

A pie chart shows how a whole is divided into parts. The full circle is 360°, and each part is a sector.

Central angle of a sector = (value ÷ total) × 360°

Example: In a school of 600 pupils, 150 like cricket. Angle = (150 ÷ 600) × 360° = 90°, which is one quarter of the circle. The sum of all central angles is 360°. If the angles are given, the value = (angle ÷ 360°) × total. A sector of 60° out of 480 pupils means (60 ÷ 360) × 480 = 80 pupils.

Percentage form: a sector of 25% has an angle of 25 × 3.6 = 90°.

Mean (arithmetic mean or average)

Mean = sum of all observations ÷ number of observations

Example: marks 8, 12, 15, 9, 6. Sum = 50, number = 5, mean = 10.

• Mean lies between the lowest and highest values.
• If every value is increased by the same number k, the mean also increases by k.
• Sum = mean × number of observations. If the mean of 6 numbers is 8, their sum is 48.
• Mean of the first n natural numbers = (n + 1) ÷ 2. For n = 9 it is 5.

Mean from a frequency table: Mean = Σ(f × x) ÷ Σf, where x is the value (or class mark for grouped data) and f is its frequency.

Median

Arrange the data in order (ascending or descending). The median is the middle value.

• Odd number of values (n): the middle one, at position (n + 1) ÷ 2.
• Even number of values: the mean of the two middle values.

Example: 3, 9, 5, 7, 11. In order: 3, 5, 7, 9, 11. Median = 7.

Example: 4, 8, 6, 10. In order: 4, 6, 8, 10. Median = (6 + 8) ÷ 2 = 7.

The median is not affected by a very large or very small value, so it is a better average when the data has extreme values.

Mode

Mode is the value that occurs the most (the highest frequency). Data can have one mode (unimodal), two modes (bimodal), more than two modes or no mode when all values occur equally often. Example: 2, 3, 3, 5, 3, 7: mode = 3. Example: 2, 4, 6: no mode. Mode is used when we want the most popular item, such as the most common shoe size.

For a moderately skewed distribution, Mode = 3 × Median − 2 × Mean (an approximate relation).

Probability

An experiment (such as tossing a coin) has outcomes. An event is one or more outcomes. Outcomes are equally likely when each has the same chance.

P(E) = number of favourable outcomes ÷ total number of outcomes

• 0 ≤ P(E) ≤ 1.
• An impossible event has probability 0. A sure (certain) event has probability 1.
• P(E) + P(not E) = 1. So P(not E) = 1 − P(E).
• The sum of the probabilities of all outcomes of an experiment is 1.

Standard experiments:

A pack of cards has 4 suits (spades, clubs, hearts, diamonds), 13 cards in each suit, 26 red and 26 black cards, 12 face cards (king, queen, jack in each suit) and 4 aces.

On a die, the prime numbers are 2, 3, 5, so P(prime) = 3/6 = 1/2. Numbers greater than 4: 5 and 6, so probability = 2/6 = 1/3.

Two coins tossed together: outcomes HH, HT, TH, TT (4 in all). P(exactly one head) = 2/4 = 1/2.

Classroom angle

Let children collect their own data: heights of classmates, favourite games or number of siblings. Making a tally table, a bar graph and then a pie chart from the same data shows why each form is used. For probability, let children toss a coin 20 times and compare the results with the expected half. Many tosses give results nearer to the expected value. Ask questions such as "which is more likely?" before introducing the formula.

Exam traps

• Bars in a histogram touch; bars in a bar graph have gaps.
• Median needs ordered data. Do not pick the middle of unordered data.
• For an even number of values, the median is the average of two middle values and may not be one of the data values.
• Mean can be a value that is not in the data; mode and median (odd count) always are data values.
• In a tally group, the fifth stroke crosses the four strokes; it is not another vertical stroke.
• Probability can never be negative or more than 1.
• A pie chart angle uses 360°, not 100.
• A die has 6 faces, but a pack of cards has 52 cards, not 56.

One-liners

• Tally marks are counted in groups of five.
• Range = highest − lowest.
• Pie chart total angle = 360°.
• Central angle = (part ÷ whole) × 360°.
• Mean = sum ÷ number of observations.
• Median of an even count = average of the two middle values.
• Mode = most frequent value.
• P(E) lies between 0 and 1.
• P(not E) = 1 − P(E).
• A fair die has 6 outcomes and a coin has 2.
• A pack has 52 cards, with 12 face cards.
• Class mark = (upper + lower limit) ÷ 2.

Practice questions

1. In tally marking, how many counts does one crossed group stand for?

   1. 5
   2. 4
   3. 6
   4. 10
   Answer

   A. 5

   Four strokes crossed by a fifth make a group of 5.

2. The total central angle of a pie chart is

   1. 180°
   2. 100°
   3. 90°
   4. 360°
   Answer

   D. 360°

   A full circle is 360°.

3. Mean of 5, 7, 9, 11 and 13 is

   1. 10
   2. 9
   3. 45
   4. 8
   Answer

   B. 9

   Sum = 45; 45 ÷ 5 = 9.

4. Median of 3, 9, 5, 7, 11 is

   1. 9
   2. 5
   3. 6
   4. 7
   Answer

   D. 7

   Ordered: 3, 5, 7, 9, 11. The middle value is 7.

5. Mode of 2, 3, 3, 5, 3, 7 is

   1. 2
   2. 5
   3. 3
   4. 7
   Answer

   C. 3

   3 occurs three times, more than any other value.

6. Median of 4, 8, 6, 10 is

   1. 6
   2. 28
   3. 8
   4. 7
   Answer

   D. 7

   Ordered: 4, 6, 8, 10. Median = (6 + 8) ÷ 2 = 7.

7. The range of 12, 7, 15, 9, 21 is

   1. 14
   2. 28
   3. 21
   4. 9
   Answer

   A. 14

   Range = 21 − 7 = 14.

8. The probability of getting a head when a fair coin is tossed is

   1. 1
   2. 1/2
   3. 1/4
   4. 2
   Answer

   B. 1/2

   1 favourable outcome out of 2 equally likely outcomes.

9. The probability of a sure event is

   1. 1
   2. 0
   3. 1/2
   4. 100
   Answer

   A. 1

   A sure event always occurs, so its probability is 1.

10. The probability of an impossible event is

    1. 1
    2. −1
    3. 0
    4. 1/2
    Answer

    C. 0

    An impossible event never occurs.

11. A die is thrown once. The probability of getting an even number is

    1. 1/6
    2. 1/2
    3. 1/3
    4. 2/3
    Answer

    B. 1/2

    Even outcomes 2, 4, 6 are 3 of 6, which is 1/2.

12. A die is thrown once. The probability of getting a prime number is

    1. 1/6
    2. 2/3
    3. 1/3
    4. 1/2
    Answer

    D. 1/2

    Primes 2, 3, 5 give 3/6 = 1/2.

13. The probability of drawing a king from a pack of 52 cards is

    1. 1/52
    2. 4/13
    3. 1/13
    4. 1/4
    Answer

    C. 1/13

    4 kings out of 52 cards is 4/52 = 1/13.

14. In a bar graph, the bars have

    1. no gaps between them
    2. equal gaps between them
    3. unequal widths
    4. different scales
    Answer

    B. equal gaps between them

    Bars have equal width and equal gaps; histograms have no gaps.

15. A graph with touching bars used for class intervals is called a

    1. histogram
    2. pictograph
    3. bar graph
    4. pie chart
    Answer

    A. histogram

    Histograms show continuous classes, so their bars touch.

16. The class mark of the class interval 10–20 is

    1. 10
    2. 30
    3. 20
    4. 15
    Answer

    D. 15

    Class mark = (10 + 20) ÷ 2 = 15.

17. In a school of 600 pupils, 150 like cricket. In a pie chart, the angle of the cricket sector is

    1. 90°
    2. 120°
    3. 60°
    4. 75°
    Answer

    A. 90°

    (150 ÷ 600) × 360° = 90°.

18. In a pie chart, a sector measures 72°. Which fraction of the whole does it show?

    1. 1/6
    2. 1/5
    3. 1/8
    4. 1/4
    Answer

    B. 1/5

    72 ÷ 360 = 1/5.

19. In a pie chart for 480 pupils, a sector of 60° shows how many pupils?

    1. 96
    2. 72
    3. 60
    4. 80
    Answer

    D. 80

    (60 ÷ 360) × 480 = 80.

20. A sector shows 25% of the data in a pie chart. Its angle is

    1. 100°
    2. 25°
    3. 90°
    4. 45°
    Answer

    C. 90°

    25% of 360° = 90°.

21. The mean of 6 numbers is 8. Their sum is

    1. 42
    2. 14
    3. 48
    4. 54
    Answer

    C. 48

    Sum = mean × number = 8 × 6 = 48.

22. The mean of the first 9 natural numbers is

    1. 5
    2. 4
    3. 4.5
    4. 9
    Answer

    A. 5

    Sum = 45; 45 ÷ 9 = 5, or (9 + 1) ÷ 2 = 5.

23. The mean of 10 numbers is 12. If each number is increased by 3, the new mean is

    1. 12
    2. 13
    3. 36
    4. 15
    Answer

    D. 15

    Adding 3 to every value adds 3 to the mean.

24. The mean of 4 numbers is 10. When a fifth number is added, the mean becomes 12. The fifth number is

    1. 22
    2. 20
    3. 12
    4. 18
    Answer

    B. 20

    New sum = 60, old sum = 40, so the fifth number is 20.

25. A bag has 3 red and 2 blue balls. One ball is drawn at random. The probability that it is red is

    1. 3/2
    2. 3/5
    3. 2/5
    4. 1/3
    Answer

    B. 3/5

    3 red out of 5 balls.

26. If P(rain) = 0.3, then the probability of no rain is

    1. 1.3
    2. 0.03
    3. 0.3
    4. 0.7
    Answer

    D. 0.7

    P(not E) = 1 − 0.3 = 0.7.

27. Two fair coins are tossed together. The probability of getting exactly one head is

    1. 3/4
    2. 1/3
    3. 1/2
    4. 1/4
    Answer

    C. 1/2

    Outcomes HT and TH are 2 of 4, which is 1/2.

28. Which average is least affected by one very large value in the data?

    1. Median
    2. Range
    3. Sum
    4. Mean
    Answer

    A. Median

    The median depends on the middle position, not on extreme values.

29. The data 2, 4, 6 has how many modes?

    1. No mode
    2. One mode
    3. Two modes
    4. Three modes
    Answer

    A. No mode

    All values occur once, so none is the most frequent.

30. How many red cards are there in a pack of 52 playing cards?

    1. 52
    2. 26
    3. 12
    4. 13
    Answer

    B. 26

    Hearts and diamonds have 13 cards each, giving 26.

31. How many face cards are there in a pack of 52 cards?

    1. 4
    2. 13
    3. 16
    4. 12
    Answer

    D. 12

    King, queen and jack in each of 4 suits give 12.

32. Which graph best compares the marks of boys and girls in each of five subjects?

    1. Tally table
    2. Histogram
    3. Double bar graph
    4. Pictograph
    Answer

    C. Double bar graph

    Double bars put the two groups side by side for each subject.

33. Which teaching activity best introduces a bar graph in Class V?

    1. Letting children collect data on their favourite games and draw bars
    2. Giving them the formula first
    3. Showing only a finished graph from the book
    4. Asking them to learn the definition
    Answer

    A. Letting children collect data on their favourite games and draw bars

    Children learn graphs best by collecting and displaying their own data.

34. A frequency table has values 2, 4, 6 with frequencies 3, 2, 5. The mean is

    1. 4
    2. 4.8
    3. 4.4
    4. 5
    Answer

    C. 4.4

    Σfx = 6 + 8 + 30 = 44; Σf = 10; mean = 4.4.

35. Statements: 1. The probability of an event can be 1.5. 2. The probability of an event can be 0. Which is/are correct?

    1. Neither 1 nor 2
    2. 1 only
    3. 2 only
    4. Both 1 and 2
    Answer

    C. 2 only

    Probability lies from 0 to 1, so it cannot exceed 1.

36. Statements: 1. Bars of a histogram touch each other. 2. Bars of a bar graph touch each other. Which is/are correct?

    1. Both 1 and 2
    2. Neither 1 nor 2
    3. 1 only
    4. 2 only
    Answer

    C. 1 only

    Bar graph bars have gaps between them.

37. Statements: 1. The median of data must always be one of the data values. 2. The mode of data must always be one of the data values. Which is/are correct?

    1. 2 only
    2. Both 1 and 2
    3. Neither 1 nor 2
    4. 1 only
    Answer

    A. 2 only

    With an even count the median is an average and may not be a data value; the mode always is.

38. Statements: 1. Mean = sum of observations ÷ number of observations. 2. Median is the most frequent observation. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    The mode, not the median, is the most frequent value.

39. Statements: 1. The probabilities of all outcomes of an experiment add up to 1. 2. P(not E) = 1 − P(E). Which is/are correct?

    1. Neither 1 nor 2
    2. 1 only
    3. 2 only
    4. Both 1 and 2
    Answer

    D. Both 1 and 2

    Both are standard rules of probability.

40. Statements: 1. The mean always lies between the smallest and the largest value. 2. A pack of cards contains 56 cards. Which is/are correct?

    1. Both 1 and 2
    2. Neither 1 nor 2
    3. 1 only
    4. 2 only
    Answer

    C. 1 only

    A pack has 52 cards.

41. Statements: 1. In a pie chart the sum of all central angles is 100°. 2. In a pie chart the sum of all central angles is 360°. Which is/are correct?

    1. 2 only
    2. Both 1 and 2
    3. Neither 1 nor 2
    4. 1 only
    Answer

    A. 2 only

    A full circle measures 360°.

42. Match the term with its meaning: P. Mean Q. Median R. Mode S. Range 1. Most frequent value 2. Middle value 3. Highest − lowest 4. Sum ÷ count

    1. P-4, Q-1, R-2, S-3
    2. P-4, Q-2, R-1, S-3
    3. P-2, Q-4, R-1, S-3
    4. P-4, Q-2, R-3, S-1
    Answer

    B. P-4, Q-2, R-1, S-3

    Mean is sum ÷ count, median is the middle, mode is the most frequent, range is the spread.

43. Match the event with its probability for one throw of a die: P. Number less than 7 Q. Number 7 R. Number 6 S. Odd number 1. 0 2. 1 3. 1/6 4. 1/2

    1. P-2, Q-1, R-4, S-3
    2. P-2, Q-3, R-1, S-4
    3. P-1, Q-2, R-3, S-4
    4. P-2, Q-1, R-3, S-4
    Answer

    D. P-2, Q-1, R-3, S-4

    Sure event 1, impossible 0, one face 1/6, odd numbers 3/6 = 1/2.

44. Match the graph with its feature: P. Pictograph Q. Bar graph R. Histogram S. Pie chart 1. Uses a key 2. Equal gaps between bars 3. Continuous classes, no gaps 4. Sectors of a circle

    1. P-1, Q-3, R-2, S-4
    2. P-1, Q-2, R-3, S-4
    3. P-2, Q-1, R-3, S-4
    4. P-1, Q-2, R-4, S-3
    Answer

    B. P-1, Q-2, R-3, S-4

    Each graph is known by this feature.

45. The marks of 7 students are 40, 35, 50, 45, 35, 60, 55. Which statement is correct?

    1. Median is 35 and mode is 45
    2. Median is 50 and mode is 35
    3. Median is 45 and mode is 50
    4. Median is 45 and mode is 35
    Answer

    D. Median is 45 and mode is 35

    Ordered: 35, 35, 40, 45, 50, 55, 60. Middle value 45; 35 occurs twice.