Mathematics lesson for Form 5 Students

Module: Descriptive : Topic: Statistics Lesson Title: Measures of Dispersion Duration: 100mins

The African Institute for Mathematical Sciences 1 Module: Chapter: Statistics Lesson: Measures of Dispersion (Variability) Class: Form 5 Additional Maths

The African Institute for Mathematical Sciences 2 Objectives of Lesson

At the end of this lesson you should be able to: • Calculate the deviations from the ; • Calculate the and of a given set; • Interpret the Value of Standard Deviation obtained.

The African Institute for Mathematical Sciences 3 Motivation In our first lesson, you learned about the mean, and for grouped and ungrouped data. At times these measures are unable to give sufficient information for worthwhile decisions to be taken about a data set. The measures of dispersion come in which says how dispersed or spread out the items of the distribution are. While the mean will just give the arithmetic average, and the mode tells you which variable is most occurring, nothing is said on how the different values are closed to the mean or to each other.

The African Institute for Mathematical Sciences 4 Material needed for the lesson For this lesson you need the following: A ruler A pencil A Pen A Calculator A4 papers or your exercise book NB: For each activity, you are expected to do it on you own before looking at the answer.

The African Institute for Mathematical Sciences 5 Plan of the Lesson

 Verification of pre-requisite;  Calculating Variance and Standard Deviation for ungrouped and grouped data;  Application exercises;  Summary;  Examples of interpretation of SD v

.

The African Institute for Mathematical Sciences 6 Pre-requisite Knowledge In the last lesson, you learned how to find Mean, Mode and Median for grouped and ungrouped data. In the nest lesson, we need the mean in the calculation of Variance then Standard Deviation.

The African Institute for Mathematical Sciences 7 Mean of Ungrouped Data

x 1 2 3 4 5 6 f 15 22 30 25 13 10

The African Institute for Mathematical Sciences 8 Mean of grouped Data

1. The marks obtained by 70 students in a test were recorded as in the table below:

Marks x 1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90

Freq f 3 5 9 14 12 9 7 6 5

a) State the modal class b) Find the mean mark correct to 1 decimal place.

The African Institute for Mathematical Sciences 9 Answer

x 1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 midmark 5.5 15.5 25.5 35.5 45.5 55.5 65.5 75.5 85.5 f 3 5 9 14 12 9 7 6 5 xf 16.5 77.5 229.5 497 546 499.5 458.5 453 427.5

The African Institute for Mathematical Sciences 10 Learning Activity 1

1. Consider a data set given as on the table below: x 2 4 5 6 7 8 9 11 f 3 6 2 1 5 6 4 1 a)Find the difference between the largest and the smallest mark.

Answer: The difference is 11 – 2 = 9. This difference is called the It gives just the difference between the highest and the lowest mark

The African Institute for Mathematical Sciences 11 Calculating Variance and Standard Deviation for ungrouped and grouped data.

The African Institute for Mathematical Sciences 12 Learning Activity 2a

1. Find the mean of the numbers 4, 5, 5, 6, 8, 9, 12, 14, 15, 16. 2. How many values are there? 3. Square the mean obtained in 1) 4. Square each of the numbers of the data set 5. Add up all the values obtained in 4). Divide this sum by the number of values obtained in 2) 6. Subtract the square of the mean from the answer obtained in 5) 7. Find the square root.

The African Institute for Mathematical Sciences 13 Solution to activity 2a

x 4 5 5 6 8 9 12 14 15 16 16 25 25 36 64 81 144 196 225 256

The African Institute for Mathematical Sciences 14 Learning Activity 2b

1. Find the mean of the numbers 4, 5, 5, 6, 8, 9, 12, 14, 15, 16. 2. Find the difference of each number from the mean (Deviations from the mean) make it a second row on a table 3. Square each of these deviations and make it a 3rd row on the table. 4. Find the mean of the sum of the squared deviations. 5. Find the square root

The African Institute for Mathematical Sciences 15 Solution to activity 2b

x 4 5 5 6 8 9 12 14 15 16

-5.4 -4.4 -4.4 -3.4 -1.4 -0.4 2.6 4.6 6.6 7.6

29.16 19.36 19.36 11.56 1.96 0.16 6.76 21.16 31.36 43.56

The African Institute for Mathematical Sciences 16 Summary of activity 2

The African Institute for Mathematical Sciences 17 Solution to Activity cont

The African Institute for Mathematical Sciences 18 Application Exercise

x 8-12 13-17 18-22 23-27 28-32 f 6 8 2 8 6

x 8-12 13-17 18-22 23-27 28-32 midpoint 10 15 20 25 30 f 6 8 2 8 6

The African Institute for Mathematical Sciences 19 Solution Cont

classes 8-12 13-17 18-22 23-27 28-32 Midpoint x 10 15 20 25 30 -10 -5 0 5 10 100 25 0 25 100 600 500 0 200 600

The African Institute for Mathematical Sciences 20 Summary

The African Institute for Mathematical Sciences 21 Interpreting the value of the TShDe Standard Deviation takes into account all the data like the mean. It is the most widely used measure of dispersion. Standard deviation looks at how spread out a group of numbers is from the mean, by looking at the square root of the variance. The variance measures the average degree to which each point differs from the mean—the average of all data points

The African Institute for Mathematical Sciences 22 Interpretation Cont.

Standard deviation can be difficult to interpret as a single number on its own. Basically, a small standard deviation that the values in a statistical data set are close to the mean of the data set, on average, and a large standard deviation means that the values in the data set are farther away from the mean, on average. The standard deviation measures how concentrated the data are around the mean; the more concentrated, the smaller the standard deviation.

The African Institute for Mathematical Sciences 23 Examples of interpretation of 1S.D In a car manufacturing industry, a certain car part has to be 3cm in diameter so as to fit properly. If during the manufacturing process, it is discovered that the SD for the diameter of this particular parts produced is large. Then the process has to stop for readjustment. Because with a large SD most of the parts will not fit properly or will not fit at all.

The African Institute for Mathematical Sciences 24 Example Cont.

2. In some situations, SD just gives information on how spread apart the data in a given data set are. In most organizations, the SD for salaries of workers might be high because we have very low salaries for cleaners and very high salaries for the PCEO.

The African Institute for Mathematical Sciences 25 Properties of Standard •DSetavnidaatrido Dneviation can never be negative; • The smallest possible value for SD is 0. This happens when every single number in the data set is exactly the same (no deviations); • The Standard Deviation is affected by extremely low and extremely high numbers in the data set; • Standard deviation has the same unit as the original data;

The African Institute for Mathematical Sciences 26 Home work 1. The gain in mass expressed in kg for 100pigs was recorded as on the table below:

Gain in kg 5-9 10-14 15-19 20-24 25-29 30-34 f 2 29 37 16 14 2 a) State the Modal class b) Find the median correct to 2 decimal places. c) Calculate the mean gain in kg correct to 2 decimal places. d) Calculate the Variance and the Standard Deviation for the data set, each correct to 2 decimal places.

The African Institute for Mathematical Sciences 27 Home work cont

2. The marks obtained by 200 students in a test were recorded as on the table below:

Marks x 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 f 10 18 20 30 49 46 20 5 2 a) State the Modal class b) How many students had less than 50? c) Find the mean mark for these students and the median mark ( 2 dp). d) Calculate the Variance and Standard Deviation for these marks. (2dp).

The African Institute for Mathematical Sciences 28 Home work Cont.

3. The ages of people at a youth club were recorded as: 14, 15, 14, 16, 14, 14, 15, 17, 15, 18, 14, 15, 15, 16, 15, 16, 15, 14, 13, 15. Find the mean and the Standard Deviation. 4. The number of eggs collected each day over a period of 21 days by a poultry owner is: 16, 17, 18, 16, 15, 18, 16, 18, 16, 15, 16, 14, 17, 16, 18, 17, 15, 17, 16, 17, 15. i) Find the modal number of eggs collected each day. ii) Find to the nearest whole number the mean number of eggs collected each day. iii) Calculate the standard deviation of the number of eggs collected each day (give answer to 2 decimal places).

The African Institute for Mathematical Sciences 29 Answers to Homework 1. a) Modal Class 15 – 19 b) Median = 17.07 correct to 2 decimal places c) Mean = 17.85 correct to 2 decimal places; d) Variance = 31.03 correct to 2 decimal places; SD = 5.57 correct to 2 decimal places 2. a) Modal Class 50 -59; b) 78 Students ; c) Mean = 51.75; Median = 53.98 d) Variance = 309.94; SD = 17.61 3. a) Mean = 15; Standard Deviation = 1.14 correct to 2 decimal places. 4. a) Mode = 16 eggs; Mean = 16 eggs, SD= 1.17

The African Institute for Mathematical Sciences 30 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 Thornes Ltd; 3. Additional Mathematics, An Integrated Core Approach, Second Edition, Piankeh Albert, Mbosso Publishers Bamenda

The African Institute for Mathematical Sciences 31 THANKS VERY MUCH AND GOOD LUCK IN YOUR STUDIES

The African Institute for Mathematical Sciences 32