Geometry, Mensuration & Statistics

Statistics

JSS 1 · JSS 2 · JSS 3

✓By the end of this lesson students will be able to collect and organise raw data using tally marks and frequency tables.
✓By the end of this lesson students will be able to represent data using pictograms and bar charts.
✓By the end of this lesson students will be able to calculate the mean, median, and mode for a given set of ungrouped data.
✓By the end of this lesson students will be able to define probability and calculate the probability of simple events.

Key concepts

Data Presentation

Data refers to a collection of facts, figures, or information. Raw data is data that has not been organised or processed. To make data easier to understand, it needs to be organised and presented clearly.

Tally Marks and Frequency Tables

Tally marks are a quick way of counting and recording data. A vertical line is used for each item, and every fifth item is represented by a diagonal line across the previous four (e.g., ||||). Frequency is the number of times an item appears in a dataset. A frequency table organises data into categories, showing the tally and the frequency for each category.

Pictograms

A pictogram (or pictograph) uses pictures or symbols to represent data. Each symbol represents a certain number of items. A key must always be provided to explain what each symbol stands for.

Bar Charts

A bar chart uses rectangular bars of equal width to represent data. The length or height of each bar is proportional to the frequency of the item it represents. There are usually gaps between the bars. Bar charts are useful for comparing different categories of data.

Averages (Measures of Central Tendency)

Averages are single values that represent a typical or central value of a set of data. The three main types of averages are the mean, median, and mode.

Mean (Arithmetic Mean)

The mean is calculated by adding up all the values in a dataset and then dividing by the total number of values. It is the most commonly used average.

Mean = (Sum of all items) / (Number of items)

Median

The median is the middle value in a dataset when the data is arranged in ascending or descending order. If there is an odd number of values, the median is the single middle value. If there is an even number of values, the median is the average (mean) of the two middle values.

Mode

The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), more than one mode (multimodal), or no mode if all values appear with the same frequency.

Probability

Probability is the measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

Basic Probability Formula

For a simple event, the probability is calculated by dividing the number of favourable outcomes (outcomes where the event happens) by the total number of possible outcomes.

P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes)

Key facts to remember

1Data is a collection of facts or information.
2A frequency table organises data using tally marks and frequencies.
3A pictogram uses symbols to represent data, with a key explaining the symbol's value.
4A bar chart uses bars of equal width to show frequencies, with gaps between bars.
5The Mean is the sum of items divided by the number of items.
6The Median is the middle value when data is arranged in order.
7The Mode is the value that appears most frequently in a dataset.
8Probability is the likelihood of an event occurring, calculated as (Number of favourable outcomes) / (Total number of possible outcomes).

Worked examples

Example 1

A JSS 2 class conducted a survey on their favourite colours. The results were: Red, Blue, Green, Red, Yellow, Blue, Red, Green, Yellow, Red, Blue, Green, Red, Yellow, Blue, Red, Green, Red, Blue, Yellow.\n(a) Organise this data into a frequency table using tally marks.\n(b) How many students chose Red as their favourite colour?\n(c) Which colour is the most popular?

I(a) First, list all the unique colours. Then, go through the data and make a tally mark for each colour. Finally, count the tally marks to get the frequency.

IIColours: Red, Blue, Green, Yellow

IIIRaw Data: Red, Blue, Green, Red, Yellow, Blue, Red, Green, Yellow, Red, Blue, Green, Red, Yellow, Blue, Red, Green, Red, Blue, Yellow.

IVFrequency Table:

V| Colour | Tally | Frequency |

VI|--------|---------|-----------|

VII| Red | |||| || | 7 |

VIII| Blue | |||| | | 5 |

9| Green | |||| | 4 |

10| Yellow | |||| | 4 |

11| Total | | 20 |

12(b) From the frequency table, the frequency for Red is 7.

13(c) The most popular colour is the one with the highest frequency. From the table, Red has the highest frequency (7).

Answer

(a) See frequency table in steps.\n(b) 7 students chose Red.\n(c) Red is the most popular colour.

Always double-check your tally counts and ensure the total frequency matches the total number of data items.

Example 2

The scores of 9 students in a Mathematics test are: 65, 70, 55, 80, 70, 60, 75, 70, 90.\nFind the:\n(a) Mean score\n(b) Median score\n(c) Mode score

I(a) To find the Mean, sum all the scores and divide by the number of students.

IISum of scores = 65 + 70 + 55 + 80 + 70 + 60 + 75 + 70 + 90 = 635

IIINumber of students = 9

IVMean = 635 / 9 = 70.56 (to 2 decimal places)

V(b) To find the Median, first arrange the scores in ascending order.

VIArranged scores: 55, 60, 65, 70, 70, 70, 75, 80, 90

VIISince there are 9 scores (an odd number), the median is the middle value. The middle value is the (9+1)/2 = 5th value.

VIIIThe 5th value in the ordered list is 70.

9(c) To find the Mode, identify the score that appears most frequently.

10In the arranged list (55, 60, 65, 70, 70, 70, 75, 80, 90), the score 70 appears 3 times, which is more than any other score.

Answer

(a) Mean score = 70.56\n(b) Median score = 70\n(c) Mode score = 70

Remember to arrange the data in order before finding the median. If there were an even number of scores, you would average the two middle scores.

Example 3

A fair die is rolled once. What is the probability of:\n(a) Rolling a 4?\n(b) Rolling an even number?\n(c) Rolling a number greater than 6?

IFirst, identify all possible outcomes when rolling a fair die. The possible outcomes are {1, 2, 3, 4, 5, 6}. So, the Total number of possible outcomes = 6.

II(a) Favourable outcome for rolling a 4 is {4}. Number of favourable outcomes = 1.

IIIP(Rolling a 4) = (Number of favourable outcomes) / (Total number of possible outcomes) = 1 / 6.

IV(b) Favourable outcomes for rolling an even number are {2, 4, 6}. Number of favourable outcomes = 3.

VP(Rolling an even number) = (Number of favourable outcomes) / (Total number of possible outcomes) = 3 / 6 = 1 / 2.

VI(c) Favourable outcomes for rolling a number greater than 6 are {}. There are no numbers greater than 6 on a standard die. Number of favourable outcomes = 0.

VIIP(Rolling a number greater than 6) = 0 / 6 = 0.

Answer

(a) P(Rolling a 4) = 1/6\n(b) P(Rolling an even number) = 1/2\n(c) P(Rolling a number greater than 6) = 0

Probability values always lie between 0 and 1, inclusive. An event with probability 0 is impossible, and an event with probability 1 is certain.

Common mistakes

✗Not arranging data in order before finding the median.
✗Confusing mean, median, and mode definitions or calculation methods.
✗Forgetting to include a key when drawing pictograms.
✗Drawing bar charts with bars touching each other (this is for histograms, not bar charts at JSS level).
✗Calculating probability with an incorrect total number of possible outcomes or favourable outcomes.

Exam tips

★Read the question carefully to determine which average (mean, median, or mode) is required.
★Always show your working steps clearly, especially for calculations of averages and probability.
★When drawing charts, use a ruler and pencil, label axes clearly, and provide a title.
★For probability questions, list all possible outcomes first to avoid errors in counting.

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