Mathematics lesson for Form 5 Students Module: Descriptive ...

Mathematics lesson for Form 5 Students

Module: Descriptive Statistics:Topic: StatisticsLesson Title: Measures of Central TendencyDuration: 100mins

The African Institute for Mathematical Sciences 1

Module: Descriptive Statistics

Chapter: StatisticsLesson: Measures of Central TendencyClass: Form 5 Additional Maths


At the end of this lesson you should be able to:

• State the mode (or modal class) for a given data set;

• Calculate the mean of a given data set;

• Find the median of a given data set.

Objectives of Lesson

3The African Institute for Mathematical Sciences

MotivationStatistics are very important for observations, analysis and mathematical prediction models. With the help of statistics we can know what happened in the past and what may occur in the future.

For example weather forecast models are built using statistics that compare prior weather conditions so as to forecast future weather conditions.

This is done by interpreting values of different measures in statistics. The first of thee measures are called Measures of Central Tendency. The three commonly used measures of central tendency are Mean, Mode and Median.


For this lesson you need the following:

A ruler

A pencil

A Pen

A Calculator

A4 papers or your exercise book

NB: For the activities and exercises, do them to the end before comparing your answer with what is in the lesson.

Material needed for the lesson


Plan of the lesson:

Verification of pre-requisite knowledge;

Mean of ungrouped and grouped data;

Mode for ungrouped and grouped data;

Median for ungrouped and grouped data.

Measures of Central Tendency


Pre-requisite KnowledgeStatistics was introduced to you in lower classes. You were taught:

• the different methods for data collection;

• quantitative and qualitative data;

• discrete and continuous variables;

• different ways of representing grouped or ungrouped data;

• how to interpret statistical data given in any form.


Qualitative and Quantitative Data

1. For the set of data below, classify them on a table having 2 columns to say whether the data is Qualitative or Quantitative:

a) Salaries of workers;

b) Number of cows owned;

c) Region of origin;

d) Political party you belong to;

e) Eye color;

f) Surface area of a plot of land;

g) Type of occupation;

h) Hours of work;

i) Sex;

j) Ages of people.


Answer

Table for the answer

Quantitative variables are measures of values or counts expressed as numbers. They are numerical variables that talk of how many, how much or how often.

Qualitative variables are measures of types and may be represented by name, symbol or a number code.


Quantitative variable Qualitative variable

Salaries of workers; Number of cows owned;Surface area of a plot of land;Hours of work;Ages of people.

Region of origin;Political party you belong to;Eye color;Type of occupation;Sex.

Discrete and Continuous Variables

2. Classify each of the variables below as discrete or continuous:

a) Number of children in a family;

b) Height of workers in a factory;

c) Number of people tested positive for COVID-19 each day;

d) Weight of children born within a given month;

e) Number of defective light bulbs produced by a factory;

f) Number of people who visit a park each week;

g) Ages of children selected for a competition;

h) Number of students doing Additional Maths ;

i) Number of cars sold by a car dealer each month;

j) Time taken to run a race.


Answer

Table for answer

Discrete variables allow only whole values to be assigned to measures while continuous variables allow for fractions or decimals to be assigned to measures


Discrete Variable Continuous Variable

Number of children in a family;Number of people tested positive for COVID-19 each day;Number of defective light bulbs produced by a factory;Number of people who visit a park each week;Number of students doing Additional Maths ; Number of cars sold by a car dealer each month.

Height of workers in a factory;Weight of children born within a given month;Ages of children selected for a competition;Time taken to run a race.

Grouped frequency table

3. Consider the table below and answer the questions that follow:

1. This is a grouped frequency distribution. True or False?

2. The variable mass is a continuous variable. True or false?

3. The number of people with masses below 72 is 88. True or False

4. The frequency for the class interval 72-75 is 25. True or False?

5. What are the class limits for the class interval 76 -79?

6. What are the class boundaries for the class 60-63:

7. What is the mid mass for the class interval 68-71?


Mass (x) in kg

60-63 64-67 68-71 72-75 76-79 80 &above

Freq (f) 8 15 40 25 10 2

Answers1. True; 2. False; 3. False; 4. True

5. Lower class limit is 76 and upper class limit is 79;

6. Lower boundary is 59.5 and Upper boundary is 63.5

7. Mid mass for the class interval 68 – 71 is 69.5


Data representation

4. Use the figure by the side to answer the questions that follow:

1.What is the name of this type of data representation ?

2.What is the percentage of mass range of 68 -71?

3.Which of the class intervals had more data?

4. Represent the above data on a bar chart


8%

15%

40%

25%

10%2%

Masses

60-63

64-67

68-71

72-75

76-79

80-

Answer

1. What is the name of this type of data representation by the side? It is a Pie chart

2.What is the percentage of mass range of 68 -71?

The percentage is 40%

3.Which of the class intervals had more data?

the interval with 40% which is the interval 68 -71.


4.4: Bar chart for the data in the pie chart above


0

5

10

15

20

25

30

35

40

45

60-63 64-67 68-71 72-75 76-79 80-

Masses

NB: If you have problems with any of the exercises, then you need to revise your notes on data collection and representation.


Measures of Central Tendency

Mean;

Mode;

Median.


Mean for ungrouped and for grouped data

Use the tables below :

1) Scores(x) for students in a test is given as on the table below:

1) The masses in kg of some students were given as on the table below:

QUESTION:

Find the mean for each of the above data set. Give answer to 2 decimal places where need be.


x 3 4 6 9 12 13

f 8 10 7 11 5 2

x 30-34 35-39 40-44 45-49 50-54

f 3 5 14 21 19

Learning Activity 1Consider the first table on marks x for students:

1. Add all the frequencies to have the total number of marks. Write down your answer.

2. Multiple each mark by the corresponding frequency. Sum the products

3. Divide the sum obtained in 2) by the sum obtained in 1)


x 3 4 6 9 12 13

f 8 10 7 11 5 2

What you will have will be

1. 8 + 10 + 7 + 11 + 5 + 2 = 43 written as σ𝑓 = 43

2. 3 × 8 + 4 × 10 + 6 × 7 + 9 × 11 + 12 × 5 + 13 × 2

= 24 + 40 + 42 + 99 + 60 + 26 = 291

⇒ σ𝑓𝑥 = 291

3. σ 𝑓𝑥

σ 𝑥=

291

43= 6.767 ≈ 6.77 to 2 decimal places

This is the mean ҧ𝑥 = 6.77 correct to 2 decimal places


Learning Activity contNow consider the second table on the masses on kg table (grouped data) :

1. Find the total frequencies

2. For each of the class intervals, find the mid-point or mid-class and make a 3rd row for the table.

3. Multiply the mid-class by the corresponding frequency to form the entries of a 4th row

4. Sum all the entries of the 4th row. That is find σ𝑥𝑓 ,𝑤ℎ𝑒𝑟𝑒 𝑥 ℎ𝑒𝑟𝑒 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑚𝑖𝑑 − 𝑐𝑙𝑎𝑠𝑠𝑒𝑠

5. Divide the total in 4) by the total in 1).


x 30-34 35-39 40-44 45-49 50-54

f 3 5 14 21 19

Answers to activities

1. σ𝑓 gives 62

2. The new table will be:

3. The new table with a 4th row will be:

4. The sum of the values in the last row of the table gives σ𝑓𝑥 = 2844

5. Find σ 𝑓𝑥

σ 𝑥=

2844

62= 45.9


x 30-34 35-39 40-44 45-49 50-54

f 3 5 14 21 19

Mid- class 32 37 42 47 52

x 30-34 35-39 40-44 45-49 50-54

f 3 5 14 21 19

Mid-class 32 37 42 47 52

xf 96 185 588 987 988

Solution continue

Therefore the mean ҧ𝑥 = 45.9

The mean is denoted by ഥ𝒙 .

The activities we have carried can be summarized as a formula as:

ҧ𝑥 =σ𝑥𝑓

σ𝑓

The mean ഥ𝒙 for a given data set is the arithmetic average of all the data available. For grouped data, the mid-classes or mid-points are considered.


Mode for ungrouped and grouped data

Consider the frequency distributions:

a)

b)

1. For table a) which is the most frequently occurring mark?

2. Still with table a) which is the mark with the highest frequency?

3. Which is the class interval in the second table with the highest frequency?


x 3 5 9 10 11 14 17

f 8 7 10 18 11 3 4

x 30-34 35-39 40-44 45-49 50-54

f 2 6 17 21 19

Solution1. The most frequently occurring mark in table a) is 10. It occurs 18

times. 10 is therefore said to be the Mode for this distribution

2. The mark with the highest frequency is 10. Its frequency is 18. 10is therefore the Mode for this distribution.

3. The class interval with the highest frequency is the class 45 – 49 with frequency of 21. The class 45-49 is therefore referred to as the Modal class for this distribution.

The Mode of a given data is the score with the highest frequency.

A data set can have two modes. For example the data 5, 4, 5, 6, 8, 6, 9, 5, 3, 6 is Bimodal. It has two modes namely 5 and 6 each occurring 3times. a data set can also be Multimodal (having many modes)


1) Find the median for the numbers 18, 24, 17, 28, 27

2) Find the Median for the data 13, 24, 32, 18, 15, 35

3-a) Find the Median for the data below:

3-b) Find the median mark for the marks in the table below:

Median for ungrouped and grouped data


x 3 4 5 6 7

f 4 6 9 5 2

x 5-14 15-24 25-34 35-44 45-54

f 5 7 19 17 7

Activity1)18, 24, 17, 28, 27a)Arrange these numbers in either ascending or descending order.b)Read the number in the middle. Since there are five numbers, the number in the middle with be in the 3rd position.

Answers:a)In ascending order we have:

17, 18, 24, 27, 28,b)The value in the third position is 24. The Median for this data is 24Descending order gives:

28, 27, 24, 18, 17The middle value is 24∴ Median = 24Here it is easy to get the middle number because there are an odd number of values

Activity:2)13, 24, 32, 18, 15, 35a)Arrange these numbers ascending order.b) Read out the two values in the middle.c)Find the average of the two numbers in the middle.Answers:

a)Arranging in ascending order gives:13, 15, 18, 24, 32, 35b)Here we have two middle values because we have an even number (6) of values.The values in the 3rd and 4th places are the middle values and are 18 and 24.

c) Their average is:18+24

2= 21

∴Median = 21


Activities for 3a)

The first table.

i) Add up all the frequencies

ii) What is the position for the middle number

iii) What is value in the middle position or if there are two of them, find their average?


Activities for 3b)

The second table

i)Add up the frequencies N;

ii)Find the position of the middle value

iii) Find the class (Median Class) where this middle value belongs.

iv) Find the lower class boundary L for this median class.

v) Write down the frequency 𝑓𝑚 of the median class f

vi) Find the cumulative frequency up the adjacent class to the median class 𝐹𝑚−1;

vii) Find the class width C for the median class

viii) Substitute in the expression: 𝐿 +1

2𝑁 − 𝐹𝑚−1

𝑓𝑚𝑐


Answer to 3a)

i) 26

ii) The middle will be between the 13th and 14th positions

iii) The number in the 13th position is 5 and that in the 14th position is 5. So their

average is: 1

25 + 5 = 5.

The median or the middle value for this distribution is 5.


Answer to 3b)

i)N is 55

ii) the median position will be 27.5

iii) The median class is 25 -34

iv) the lower class boundary is 24.5

v) The frequency of the median class 𝑓𝑚 = 19

vi) the cumulative frequency for the adjacent lower class 𝐹𝑚−1= 12

vii) the class width C = 34.5 -24.5 = 10

viii) Substituting in the expression 𝐿 +1

2𝑁 − 𝐹𝑚−1

𝑓𝑚𝑐 give the value of the

expression to be 32.7.

This value is the Median for the above data so we have

Therefore Median = 24.5 + 27.5−12

31 −12(34.5 − 24.5) = 32.7

⇒ 𝑚𝑒𝑑𝑖𝑎𝑛 = 24.515.5

1910 = 32.7


Summary

The formula used above is: 𝑀𝑒𝑑𝑖𝑎𝑛 = 𝐿 +1

2𝑁 − 𝐹𝑚−1

𝑓𝑚𝑐 :

Where L is the lower class boundary of the median Class. In our case the class is 25 – 34 and the lower boundary is 24.5;

N is the total number of observations and is 55 such that half of it is 27.5

𝐹𝑚−1 is the cumulative frequency for the adjacent lower class;

𝑓𝑚 is the frequency of the median class which is 19 for the above data.

C is the class width which for our data is 34.5 – 24.5 = 10


Solution continue

This median could also be determined by using a cumulative frequency curve. Here we plot the cumulative frequencies against the upper class boundaries (could still use the upper class limit)

Cumulative Frequency Table for the above data is:

The cumulative frequency curve (Ogive) is on the nest slide. The mark indicated by the Red line gives the Median


Masses <14.5 24.5 <34.5 <44.5 <54.5

CF 5 12 31 42 55

Cumulative Frequency curve


Cumulative Frequency curve

Med

-100

-100

0 100

100

Marks

CF

Application Exercises1) State the mode of the numbers 5, 7, 12, 12, 12, 17, 17, 19,21,

2) The height of 40 trees were measured and the data recorded as:

What is the modal class ?

3) The weight in kg of parcels delivered by DHL were recorded as 7.4; 8.2; 11.1; 7.8; 2.5; 5.6; 7.1; 8.9; 2.3; 2.7; 2.9; 4.1; 5.3. Find the Median height.


Height in m 2-4 4-6 6-8 8-10 10-12

Frequency 2 7 11 13 7

Application Exercises Cont.

4) 30 seedlings were measured and their heights in cm given as on the table below:

a) State the modal class

b) Find the mean height of the seedlings


Height (cm) 5-15 15-25 25-35 35-45 45-55


Answers to above application exercises

1) The mode is 12. It the most occurring number with a frequency of 3.

2) The modal class is 8 – 10 with a frequency of 13.

3) 7.4; 8.2; 11.1; 7.8; 2.5; 5.6; 7.1; 8.9; 2.3; 2.7; 2.9; 4.1; 5.3. Rearranging gives:

2.3; 2.5; 2.7; 2.9; 4.1; 5.3; 5.6; 7.1; 7.4; 7.8. 8.2; 8.9; 11.1

There are 13 values so the middle value is in the 7th position

Therefore the Median = 5.6.

4a) The modal class is 25 -35


4b) The mean of the grouped data

𝑀𝑒𝑎𝑛 ҧ𝑥 =σ 𝑥𝑓

σ 𝑓= 810

30= 27

∴ ҧ𝑥 = 27𝑐𝑚


Height 5-15 15-25 25-35 35-45 45-55

Mid class 10 20 30 40 50


Mid-class ×frequency

60 80 450 120 100

The use of Mean, mode and median

The three measures are useful in different context. For example:

An employer considering the production capacity of workers, will use Mean of past production or past performance;

A shop keeper wanting to keep minimum stock will use the Mode as this will inform on what is most likely to sell;

For a union wage navigator, the median will be used so that the few high earners would not affect the average of the wages paid.

These three measures are also referred to as Measures of Location.


Summary The mean, median and mode make data meaningful. Each has a specific way of looking at for a given data, as each one can give you a different insight into the way information behaves in the real world around you.

The mode tells you what most of the pieces of data are doing within a set of information.

The most common usage of mean is In school. The final grade you get in a course or at the end of a term is usually a mean.

The median number in a group refers to the point where half the numbers are above and the other half are below it.

Using these measures in everyday life involves not only understanding the differences between them, but also which one is appropriate for a given situation


Homework1. The mean of five numbers is 15 and the mean of 8 other numbers is 2.

Find the mean of the thirteen numbers.

2. Find the mean, mode and median for this set of numbers: 3,3,3,7,7,7,7,7,8,8,8,8,8,8,10,10,10,12,12,24.

3. At a college the ages of 50 students were recoded as on the table below:

a) State the Modal age of the students

b) Calculate the mean age of the students

c) Find the median age using any method you know.

4. Given that the mean of the numbers 0, 1, 7, 1, 1, 7, 2, x, x, y , where x and y are integers, is 4. Show that 2x + y = 21


X years 16-20 21-25 26-30 31-35 36-40 41-45 46-50 51-55

f 5 10 15 9 5 3 2 1

Answers to Homework

1. Mean of the 13 numbers will be 7.

2. Mean = 8.5; Mode = 8; Median = 8

3. Modal Class = 26 – 30; Mean 30.1; Median = 28.83

4. 19 + 2x + y = 40 ⇒ 2𝑥 + 𝑦 = 19


References

1. Integrated Core Approach, Ordinary Level Mathematics, Third Edition, Piankeh Albert, Mbosso Publishers Bamenda;

2. Mathematics 9, M.J. Tipler, J Douglas (2004),Nelson ThornesLtd;

3. Additional Mathematics, An Integrated Core Approach, Second Edition, Piankeh Albert, Mbosso Publishers Bamenda


The next lesson will be on Measures of dispersion

THANKS VERY MUCH AND GOOD LUCK IN YOUR STUDIES


END OF LESSON


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