StatisticsMathematics 8th Grade

Chapter 1 Unit 5

Module Objectives

1. In this course , you will learn , what is Data , Observation , Range, frequency, Class interval,Exclusive and Inclusive Class intervals, Mid points of Class interval.

2. Construct a frequency distribution table for inclusive and Exclusive Class Interval3. Draw a histogram for the given frequency distribution table.

1.5.1 Introduction• Statistics is a mathematical science related to the collection , analysis , interpretation and

presentation of Data.

• Used in various fields such as Weather , business , Education , research,etc.

WHAT IS DATA ? ? A collection of Numerical facts which give particular information is called DATA

Example

Marks obtained by 10 , 8th Standard Students in Mid Term Exam: This collection of Numerical enteries is called Observation

56,30,44,56,78,35,78,46,88,46• Such collection of Data is called RAW DATA .• This Data can be arranged in Ascending / Descending Order

Let us arrange it in Descending order : 88,78,78,56,56,46,46,44,35,30• Highest Score – 88• Lowest Score – 30

What is RANGE ? Range is the difference between the highest and lowest scores.In this example the RANGE is 88-30 =58

Introduction. Contd.

Example

Marks obtained by 10 , 8th Standard Students in Mid Term Exam: 56,30,44,56,79,35,78,46,56,46

In the above Example , the Value 46 is repeated 2 times and value 56 Is repeated 3 times.

The Number of times a particular observation (score) occurs in a data is called is called its FREQUENCY

The Representation of this data in a Tabular Format is called as a FREQUENCY DISTRIBUTION TABLE, where tallies are used to mark the counts I I I represents 3 and I I I I I represents 5.

Constructing a Frequency Distribution Table

Example 1The Marks scored by 20 Students in a Unit Test out of 25 are given below12,10,08,12,04,15,18,23,18,16,16,12,23,18,12,05,16,16,12,20Prepare a Frequency Distribution Table for the same.

MARKS TALLY MARKS No. of Students (FREQUENCY)

23 I I 2

20 I 1

18 I I I 3

16 I I I I 4

15 I 1

12 I I I I I 5

10 I 1

08 I 1

05 I 1

04 I 1

TOTAL 20 20

Another Example of a Table

1.5.2 GROUPING DATAOrganising the data in the form of frequency distribution table is called grouped frequency distribution of Raw data.

What do we do when we have large data ? ? ?

Example 2 : Consider the following marks (out of 50) scored in Mathematics by 50 Students of 8th Class.

41,31,33,32,28,31,21,10,30,22,33,37,12,05,08,15,39,26,41,46,34,22,0911,16,22,25,29,31,39,23,31,21,45,47,30,22,17,36,18,20,22,44,16,24,1027,39,28,17

Groups

Tally Marks Frequency

0-9 III 03

10-19 IIIII , IIIII 10

20-29 IIIII,IIIII,IIIII,I 16

30-39 IIIII,IIIII,IIIII 15

40-49 IIIII , I 06

50-59 0

Total 50 50

For convenience we make groups of observations like 0-9, 10-19 and so on. We obtain a frequency of distribution of the no. of observations coming under each groupThe data presented in this manner is said to be grouped and the distribution obtained is called Grouped Frequency Distribution.

From this , we can notice that .

a.Max no. of students have scored between 20-29.b.Only 3 students have scored less than 10 Marks.c.No Student has scored 50 or more than 50.

1.5.2 GROUPING DATA, Contd.

Groups

Tally Marks Frequency

0-9 III 03

10-19 IIIII , IIIII 10

20-29 IIIII,IIIII,IIIII,I 16

30-39 IIIII,IIIII,IIIII 15

40-49 IIIII , I 06

50-59 0

Total 50 50

In the table beside, Marks are grouped into 0-9, 10-19, etc. Each of these groups is called as a Class Interval or Class.This method of Grouping is called Inclusive method.

Class Limit: In the class interval, say (10-19) , 9.5 is called as lower class limit and 19.5 is called the upper class limit

Note: Class limit in Inclusive method.

Lower class limit - Subtract 0.5 from the lower score Upper Class limit - Add 0.5 to the upper score

Class Size: The no. of scores in the class interval say (10-19) including 10 and 19 is called the class size or width of the class. Class Size (10-19) = 10

Class Mark : the midpoint of a class is called its class mark (or midpoint of class interval)Ex. Class mark of (10-19) is (10+19)/2=14.5Class mark of (10-20) is (10+20)/2=15

Grouping Data Contd.

Groups

Tally Marks Frequency

0-10 III 03

10-20 IIIII , IIIII 10

20-30 IIIII,IIIII,IIIII,III

18

30-40 IIIII,IIIII,III 13

40-50 IIIII , I 06

Total 50 50

The data in the previous example can also be shown this way, like class intervals 0-10, 10-20 and so on.

Here , observe that 10 occurs in both classes ,(0-10) and (10-20). But we cannot have 10 in both classes simultaneously. To avoid this , we must follow a convention that the 10 will belong to a higher class here (10-20) and not (0-10).Similarly , 30 must belong to (30-40) and not (20-30).This method of grouping data is called the Exclusive method.

Class Limit : In the class interval (10-20) 10 is called the lower limit and 20 is called the upper limit.Class Size : The difference between the upper limit and the lower limit is called the class size / width. The width of the class (10-20) is 20-10=10

1.5.3 Histogram

• A Histogram is a representation of a frequency ditribution table by means of rectangles whose widths represent class intervals and whose areas are proportional to the corresponding fre quencies. If the length of all class intervals are same , then the frequency is proportional to the height of the rectangle.

Note : Histograms are created only for class intervals which are exclusive. If the class intervals are inclusive , then they must be converted to exclusive, by applying correction factor.

Work Sheet

How to draw a histogram for a given Frequency Distribution Table ?

Class Interval

Frequency

0-9 5

10-19 8

20-29 12

30-39 18

40-49 22

50-59 10

The given distribution is of the inclusive form. It has to be converted to exclusive form by applying a correction factor d/2.Where d=(lower limit of a class - upper limit of a class before it)

Here we have Actual upper limit = stated limit +d/2Actual lower limit = stated limit –d/2

Consider the class limit 10-19. You get d=lower limit of the class –upper limit of a class before it) = 10-9 = 1Hence d=1 or d/2=0.5

Actual Upper Limit = Stated limit + d/2 =19+0.5 =19.5Actual Lower Limit = Stated limit - d/2 =10-0.5 = 9.5

Converting the previous table into Exclusive format

Stated Class Interval

Actual Class Interval

Frequency

0-9 -0.5-9.5 5

10-19 9.5-19.5 8

20-29 19.5-29.5 12

30-39 29.5-39.5 18

40-49 39.5-49.5 22

50-59 49.5-59.5 10

To Plot the graph

Draw x axis and y axisChoose appropriate scale. 1 cm =10 on x axis and 1 cm =6 on y axis

Mark class intervals on X axis and frequency on Y axis

Plot the graph as shown for all class intervals.

Note : the height of the rectangles represent the frequencies and the base represents class intervals.You can ignore the spaces in between the rectangles , so that this becomes a continuous distribution.

References

Youtube Links

https://www.youtube.com/watch?v=DXj4Q0jhLsI

https://www.youtube.com/watch?v=VGmfTJhv-i0

Module Objectives

• Mean

• Median

• Mode

Mean, Median & Mode• 3 important quantities associated with

statistical data.• Give clear picture of behavior of an

experiment.• Called ‘measures of central tendencies’.

Mean• Average of collected data.• Gives an idea of how the experiment is behaving.• Example:

• Scores of student of a class in Maths test is given below:

59, 46, 77, 92, 64, 98, 25, 18, 44, 22Mean = Sum of all values / Number of values = (59+46+77+92+64+98+25+18+44+22)/10 = 54.5From the mean value, we can infer that the students of

this class have an average performance.

Mean for an Ungrouped dataFormula:Mean = (sum of all values of observation)

(the number of observations) If x1, x2, x3 . . . xN are the values of N observations, then,Mean = x1 + x2 + x3 + . . . . . . + xN

N

Notations• Sum is denoted by ∑ and read as sigma.• Mean is denoted by X and read as X-bar.• Hence we have,

X = ∑x

N

Mean - Examples• Example 1: Find the mean of first six even natural

numbers.Solution: The first six even natural numbers are 2, 4, 6,

8, 10, 12. There are six values or observations.Therefore N = 6. The observations arex1 = 2, x2 = 4, x3 = 6, x4 = 8, x5 = 10, x6 = 12Hence ∑x = 2 + 4 + 6 + 8 + 10 + 12 = 42Mean is given by,

X = ∑x = 42 = 7

N 6

Mean - Examples• Example 2: A football team had drives of 43,

42, 45, 44, 45, and 48 yards. Find the mean drive for the team.

Here, N = 5∑x = 43 + 42 + 45 + 44 + 45 + 48 = 117Hence mean is given by

X = ∑x = 117 = 23.4

N 6

Mean for grouped data

• If a large number of values are given, then calculating mean using the above method becomes difficult.

• In these cases, we group the data and prepare a frequency distribution table. From the frequency distribution table, we can find mean.

Mean for grouped dataExample 1

• Example 1: The number of goals scored by a hockey team in 20 matches is given here.

4, 6, 3, 2, 2, 4, 1, 5, 3, 0, 4, 5, 4, 5, 4, 0, 4, 3, 6, 4Solution:Follow the below steps:

Mean for grouped dataExample 1

Step 1: Prepare a frequency distribution table.

Mean for grouped dataExample 1

Step 2: Calculate fx for each score by multiplying f and x. Add all fx to get ∑fx.

Scores(x)

Frequency(x)

fx

0 2 0

1 1 1

2 2 4

3 3 9

4 7 28

5 3 15

6 2 12

N = 20 ∑fx = 69

Mean for grouped dataExample 1

• Step 3: Now the mean is calculated as X = sum of the scores

number of scores = ∑fx N = 69 20

Mean for grouped dataExample 2

• Find the mean of given frequency table.

Class - Interval Frequency

0 – 4 3

5 – 9 5

10 – 14 7

15 – 19 4

20 – 24 6

N = 25

Mean for grouped dataExample 2

• Solution:• Step 1: Find the mid point of each class

interval.• Step 2: Calculate fx by multiplying the values

of f and x.• Step 3: Add all fx and calculate ∑fx.

Mean for grouped dataExample 2

Class - Interval Midpoint of Class Interval (x)

Frequency(f)

fx

0 – 4 2 3 6

5 – 9 7 5 35

10 – 14 12 7 84

15 – 19 17 4 68

20 – 24 22 6 132

N = 25 ∑fx = 325

Mean for grouped dataExample 2

• Step 4: Now the mean is calculated as X = sum of the scores

total number of scores = ∑fx N = 325 = 13 25

Median

• It is the mid-point of the data after being arranged in asceding or descending order.

• Example:8, 5, 3 , 5, 6, 10, 7Arranging in ascending order,3, 5, 5, 6, 7, 8, 10

• Hence median is3 5 5 6 7 8 10

Median for an ungrouped data

Step 1: Arrange the given scores in ascending or descending order.Step 2: Check if the number of scores is odd or even.If it is odd, median is the middle most score in the arranged set.If it is even, median is calculated as average of 2 middle most scores.

Median for an ungrouped dataExample 1

Example 1: Find the median of the data:26, 31, 33, 37, 43, 8, 26, 33.Solution: Arranging the score in ascending order,26, 31, 33, 37, 38, 42, 43Here the number of terms is 7. The middle term is 4th one and it is 37.Hence, median = 37

Median for an ungrouped dataExample 2

Example 2: Find the median of the data:32, 30, 28, 31, 22, 26, 27, 21.Solution: Arranging the score in ascending order,21, 22, 26, 27, 28, 30, 31, 32Here the number of terms is 8. The median is the average of two middle terms which are 27 and 28.Hence, median = (27 + 28) / 2 = 27.5

Median for an ungrouped data

Steps to calculate median for any given N values.Step 1: Arrange the values in ascending or descending order.Step 2: If N is odd, thenMedian is at (N + 1)/2 –th placeIf N is even, thenMedian is at ½(score at N/2 –th place + score at (N/2 + 1) –th place).

Median for grouped dataExample

Example 1: Find the median for the following grouped data.

Class Interval Frequency

1 – 5 4

6 – 10 3

11 – 15 6

16 – 20 5

21 – 25 2

N = 20

Median for grouped dataExample

Solution:Step 1: Value of N = 20 (Even). Hence there are 2 middle scores – 10th score and 11th score.Step 2: Calculate Cumulative frequency as show in the table.

Median for grouped dataExample

Calculating cumulative frequency:Class Interval Frequency Cumulative

Frequency (fc)

1 – 5 4 4 4

6 – 10 3 7 4 + 3 =7

11 – 15 6 13 7 + 6 = 13

16 – 20 5 18 13 + 5 = 18

21 – 25 2 20 8 + 2 =20

N = 20

Median for grouped dataExample

Solution:Step 3: Find median class. Count frequencies from first class interval and find the class interval that 10th score lies in. We find that it is in class interval (11-15), called median class. Corresponding frequency is 6.Step 4: Find lower real limit (LRL) of median class. Here it is 10.5 and cumulative frequency above this class is 7.

Median for grouped dataExample

Solution:So we have,LRL = 10.5frequency of median class (fm) = 6

cumulative frequency of median class (fc) = 7

size of class interval (i) = 5

Median for grouped dataExample

Solution:Step 5: Calculate median using the formula median = LRL + ((N/2) – fc) x i

fm

median = 10.5 + (20/2) – 76

= 13

Mode

• It is the score that occurs frequently in a given set of scores.

Mode for an ungrouped data

Example 1: Find the mode of the data:15, 20, 22, 25, 30, 20, 15, 20, 12, 20Solution:Here 20 appears maximum times (4 times).Hence mode = 20

Mode for an ungrouped data

Example 2: Find the mode of the data:5, 2, 3, 3, 5, 7, 6, 3, 4, 3, 5, 8, 5Solution:Here 3 and 5 appear 4 times.Hence mode = 3 and 5

Mode for grouped data

Example: Find the mode of the data:

Solution:Here the maximum frequency is 22. The number corresponding to maximum frequency is the mode. Hence mode = 15

Number 12 13 14 15 16

Frequency 7 9 6 22 20

References

• https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-data-statistics/cc-6th-statistics/v/mean-median-and-mode

• https://www.youtube.com/watch?v=uhxtUt_-GyM