Statistics & Probability

STATISTICS

By: kasskassmaths​

​General Mathematics Syllabus

Section #5 , Objectives ​1 - 14

Table of Contents

Definitions - Slide 3

Frequency - Slide 4

​Class Features - Slide 6

​​Measures of central tendency (Mean, Mode, Median) - Slide ​8

​Measures of Dispersion - Slide ​11

​Statistical Diagrams - Slide​ 18

Probability - Slide 34

​Statistics & Probability from a Cumulative Frequency Curve - Slide

​​

Past Paper Questions to Practice - Slide​

Definitions

1.Population - the total amount of things, persons or objects under study

​

2.Sample - A representative of the population. A part or subset of the population.

3.Data ​- a collection of numerical or non-numerical facts

​

4.Discrete Data / Discrete Variables - this is data that has exact values. / a variable which takes a certain definite value. E.g. cash, shoe sizes, no. of siblings

​

5.Continuous Data / Continuous Variables - ​a variable is said to be continuous if it is approximated at different levels of precision. / a variable which can take any value within a given range an can be obtained by measurement. E.g. height, weight, mass, area, volume

​​6. A parameter is a characteristic of a population which describes the population.

Frequency

The Frequency of an event is defined as the number of times an event has occurred. ​
A major way of organizing raw grouped and ungrouped data is using tally and a frequency table. tally - 4 vertical strokes and a horizontal one through it representing 5

​

​1) Frequency table for ungrouped data

​E.g. There are 50 participants in a competition. The score
of each person is listed.
Note the frequency table drawn for the scores.

2) Frequency Table for ungrouped data.

Sometimes the data under consideration has such a large range of values that it is most useful to collect these values into groups or classes. Other times, the data given is better analyzed when grouped rather than ungrouped.​

Class Features

Measures of central tendency (Mean, Mode, Median)

​E.g. Find the mean, mode, and median of the following values which shows heights of students in cm.
115 120 163 163 163 167 168 169 190 192

​

E.G.

​

Measures of Dispersion

They are :

1.Range - difference between the largest and smallest observations / values in a sample

2. Interquartile range (I.Q.R) - the difference between the upper and lower quartiles of a distribution

3. Semi-Interquartile Range (S.I.Q.R) - half the the difference between the upper and lower quartiles of a distribution

4. Standard Deviation.

1. ​Range

​Ungrouped Data :

Range = largest observation - smallest observation

E.g. Calculate the range of the basic wages ​of a factory worker as seen below.

$175 $160 $196 $149 $185 $167 $148​

Range = $195 - $148 = $47​

​

Grouped Data :

​Range = upper boundary limit of the largest observation - lower boundary limit of the smallest observation.

E.g. 32 children were asked to estimate the length of a metal rod to the nearest centimetre. The table shows the results obtained. ​

​

L.B.L = 34.5cm & U.B.L = 42.5cm

Range = 42.5 - 34.5

= 8cm ​

​Estimated length (cm)	Frequency ​
35​	1​
36​	3​
37​	4​
38​	8​
39​	6​
40​	5​
41​	3​
​42	2​

2.Interquartile Range (I.Q.R)

e.g.

100 students wrote a test in which the maximum mark that could be obtained was 5. The mark of each student is listed in the frequency table below. Find the Interquartile range and the semi-quartile range.

​Mark	Frequency ​	​cumulative frequency
​1	30​	0​ - 30
2​	26​	31 - 56
3​	20​	57 - 76
4​	14​	77 - 90
5​	10​	91 - 100​

4. Standard Deviation

​

Standard deviation is a measure of how dispersed the data is in relation to the mean. Low/ Small standard deviation means data are clustered around the mean (there is less spread), and high/large standard deviation indicates data are more spread out about the mean. When comparing 2 or more sets of data, the lower S.D means less spread and the Larger S.D means more spread.

​Note : You don't need to know how to calculate this at this level.​

E.G. June 2022 P2 5(b)

​In class B the mean height of the students is 123.5cm, and the standard deviation is 29.87. For class A the standard deviation is 21.38. Using the information provided and your response from (a), comment on the distribution of the heights of the students in Class A and B.

​Solution : ​The mean height of the students in Class B (123.5cm) is approximately the same as that of class A (122.2cm). However the standard deviation of the score in Class B (29.87) is significantly higher than that in Class A (21.38). Standard Deviation is a measure of the spread of data. This suggests that the heights of students in Class B are more spread out while those in Class A are more clustered around the mean.

Statistical Diagrams

1) Pie Chart - this is a circular diagram used to represent statistical information. The circle is divided into sectors based on given data. Before constructing a pie chart, the sector angles have to be calculated. Recall that a circle contain 360 degrees. Each sector or area is directly proportional to the magnitude of the information that it is representing. Note : geometry tools must be used ​when drawing the pie chart.

​E.g. The table shows the number of graduates by subject from a school. Construct a pie chart of radius 2.5cm to represent this information.

​subject	​# of grads
Maths​	17​
English ​	25​
Physics​	10​
Chemistry ​	15​
​History	23​

NOTE

​When drawing the following graphs, use appropriate scales on the x and y axes based on the data that will be plotted. x axis = horizontal , y axis = vertical. Such as

1 small block = 1 unit of the observation / variable etc

​1 big block = 1 unit of the observation / variable etc

​1 big block = 10 units of the observation / variable etc

​

Note :

The both axes do not have to have the same scale unless a question specifies so.

Label axes​ based on data of the question. (frequency, height, points etc)

​Observe any patterns or trends the graphs may posses. (downward, upward, constant etc)

​

2) ​Bar Chart / Column graphs - this consists of a number of rectangular bars of the same width which can be drawn vertically or horizontally and are evenly spaced out. ​The height or length of each bar is directly proportional to the magnitude of the data it is representing. This type of graph / chart is usually drawn on a graph sheet where an appropriate scale is used and the both axis' has a purpose.

​

​E.g. The table below shows the height of 7 waterfalls. See the bar chart (vertical) representing this.

​Name	Location	​​Height(cm)
​A	East ​	979​
B​	West ​	610​
C​	North ​	488​
D​	South ​	457​
E​	North East ​	404​
F​	South East ​	300​
G​	North West​	251​

3) ​Line Graph - this shows the data by means of drawing a line as the name suggests. It is constructed by joining a set of points together in a consecutive manner. They are very good for showing upward and downward trends.

This graph is drawn on graph sheets​

​E.g. The table below represents the quantity of fruits grown annually on a farm. Draw a line graph to represent this and determine which period was there a decrease in the quantity of fruits.

​

​YEARS	FRUITS (TONNES)​
1986​	50​
1987​	200​
1988​	300​
1989​	350​
1990​	450​
1991​	200​
1992​	500​
​1993	600​

1990 - 1991, there was a decrease in the quantity of fruits.

4)​ Histogram - it consists of a number of rectangular bars that are always drawn vertically, joined side to side without leaving any space. No bar = 0 frequency of that specific data. Frequency is always plotted on the vertical axis and the horizontal axis contains the variable or observation. The frequency of an observation is directly proportional to the height of each bar. The width is the same for each bar.

​Note : For Ungrouped Data, either the class boundaries or class mid-points is plotted along the horizontal axis against the corresponding frequencies.

​

​These graphs are drawn on graphs sheets.

E.g. Ungrouped Data : The frequency table below shows the number of points gained by the teams in a series of cricket matches. Draw a histogram to represent the data.

​No. of Points	Frequency ​
0​	3​
1​	2​
2​	0​
3​	5​
4​	6​
5​	7​
6​	2​
7​	1​
8​	0​
9​	4​
​10	1​

E.g. Grouped Data : The weights of 100 students correct to the nearest kilogram are given in the frequency tale below. Draw a histogram to represent the information given above.

​class (kg)	frequency ​	​midpoint
35 -​ 39	5​	37​
40 -​ 44	7​	42​
45 -​ 49	8​	47​
50 -​ 54	10​	52​
55​ - 59	13​	57​
60​ - 64	17​	62​
65​ - 69	15​	67​
70​ - 74	12​	72​
75​ - 79	9​	77​
​90 - 84	4​	82​

​5) Frequency Polygons - this is like a line graph but it indicates the spread of a given distribution. The frequency polygon for ungrouped data is obtained by plotting the observation or variable against the corresponding frequency and the drawing straight lines in order to join consecutive points. The frequency polygon for grouped data is obtained by plotting the mid-points of the class intervals against the corresponding frequencies and drawing straight lines in order to join consecutive points. These are drawn on graph sheets.

E.g. Ungrouped Data : The marks obtained by 50 students in a test were collected in the table. Draw a frequency polygon representing the data in the table.

​Marks	Frequency ​
1​	1​
2​	2​
3​	4​
4​	7​
5​	9​
6​	10​
7​	8​
8​	5​
9​	3​
​10	1​

E.g Grouped Data : Given the following table, construct a frequency polygon using the midpoint.

​

​Marks	Frequency​	Midpoint ​
1 - 10​	3​	5.5​
11 - 20 ​	3​	15.5​
21 - 30​	8​	25.5​
31 - 40 ​	15​	35.5​
41 - 50​	​19	45.5​

Probability

​E.x. A fair six-sided die is rolled once. ​Let :

"A" is the event "the score is even."

"B" is the event "the score is odd."

"C" is the event "the score is at least 5."

"D" is the event "the score is a factor 60."

"E"​ is the event "the score is greater than 10."

​The sample space is S = {1, 2, 3, 4, 5, 6}

A = {2,4,6} B = {1, 3, 5} C = {5, 6}

D = {1, 2,3,4,5,6}​

​E = {}

​A pack/deck of playing cards contain :

​4 Ace , 4 Jack, 4 Queen, 4 King, 36 number cards from 2 to 10, 9 clubs, 9 spades, 9 hearts, 9 diamonds. Total 52​ cards.

Contingency Table

​	​1	2​	​3	​4	5​	6​	7​	8​
​coin 1	H​	H​	H​	T​	H​	T​	T​	T​
coin 2​	H​	H​	T​	H​	T​	H​	T​	T​
coin 3​	​H	T​	H​	H​	T​	T​	H​	T​

​Statistics & Probability from a Cumulative Frequency Curve

​1) The table below shows the amount of money spent by 40 students for a week at the school cafe.

(a) Complete the table showing lower and upper boundaries (U.C.B. & L.C.B.), cumulative frequency and an additional column to show the points to be plotted.

(b)Plot a graph to display this information.

(c)

i) Determine the median amount of money spent to the nearest dollar.

ii) What is the probability that a student spends less than $23 in a week.

iii) Using the graph, calculate the Interquartile range ​and semi-interquartile range.

​Scale :

​x axis - 1cm = 5.25

y axis - 1cm = 2.5​

​(C) i) Median / Q₂= 20 (y axis)

Respective x = 31.55

= $ 32 ​

​

​(iii) IQR = Q₃ - Q₁

= 40.5 - 20.5

= $20 ​

​

Past Paper Questions to Practice

June 2019 No 5 (a) - (e)​

July 2021 No. 5 (a) & (b)

May / June 202​2 No. 5 (a) - (c)
May / June 2023 No. 5
May / June 2024 No. 5