1. Statistics Mathematics 8th Grade Chapter 1 Unit 5

2. 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 Interval 3.Draw a histogram for the given frequency distribution table.

3. 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

4. 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.

5. Constructing a Frequency Distribution Table Example 1 The Marks scored by 20 Students in a Unit Test out of 25 are given below 12,10,08,12,04,15,18,23,18,16,16,12,23,18,12,05,16, 16,12,20 Prepare a Frequency Distribution Table for the same. MARKSTALLY MARKSNo. of Students (FREQUENCY) 23I 2 20I1 18I I I3 16I I 4 15I1 12I I I I I5 10I1 08I1 05I1 04I1 TOTAL20 Another Example of a Table

6. 1.5.2 GROUPING DATA Organising 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. 41,31,33,32,28,31,21,10,30,22,33,37,12,05,08,15,39,26,41,46,34,22,09 11,16,22,25,29,31,39,23,31,21,45,47,30,22,17,36,18,20,22,44,16,24,10 27,39,28,17 Group s Tally MarksFre que ncy 0-9III03 10-19IIIII, IIIII10 20-29IIIII,IIIII,IIIII,I16 30-39IIIII,IIIII,IIIII15 40-49IIIII, I06 50-590 Total50 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 group The 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.

7. 1.5.2 GROUPING DATA, Contd. Group s Tally MarksFre que ncy 0-9III03 10-19IIIII, IIIII10 20-29IIIII,IIIII,IIIII,I16 30-39IIIII,IIIII,IIIII15 40-49IIIII, I06 50-590 Total50 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=14.5 Class mark of (10-20) is (10+20)/2=15

8. Grouping Data Contd. Group s Tally MarksFre que ncy 0-10III03 10-20IIIII, IIIII10 20-30IIIII,IIIII,IIIII,I II 18 30-40IIIII,IIIII,III13 40-50IIIII, I06 Total50 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

9. 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 frequencies. 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.

10. Work Sheet How to draw a histogram for a given Frequency Distribution Table ? Class Interval Frequenc y 0-95 10-198 20-2912 30-3918 40-4922 50-5910 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/2 Actual 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 = 1 Hence d=1 or d/2=0.5 Actual Upper Limit = Stated limit + d/2 =19+0.5 =19.5 Actual Lower Limit = Stated limit - d/2 =10-0.5 = 9.5

11. Converting the previous table into Exclusive format Stated Class Interval Actual Class Interval Frequenc y 0-9-0.5-9.55 10-199.5-19.58 20-2919.5-29.512 30-3929.5-39.518 40-4939.5-49.522 50-5949.5-59.510 To Plot the graph Draw x axis and y axis Choose 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.

12. References Youtube Links https://www.youtube.com/watch?v=DXj4Q0jhLsI https://www.youtube.com/watch?v=VGmfTJhv-i0

13. Module Objectives Mean Median Mode

14. Mean, Median & Mode 3 important quantities associated with statistical data. Give clear picture of behavior of an experiment. Called ‘measures of central tendencies’.

15. 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, 22 Mean = Sum of all values / Number of values = (59+46+77+92+64+98+25+18+44+22)/10 = 54.5 From the mean value, we can infer that the students of this class have an average performance.

16. Mean for an Ungrouped data Formula: Mean = (sum of all values of observation) (the number of observations) If x 1, x 2, x 3... x N are the values of N observations, then, Mean = x 1 + x 2 + x 3 +...... + x N N

17. 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

18. 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 are x1 = 2, x2 = 4, x3 = 6, x4 = 8, x5 = 10, x6 = 12 Hence ∑x = 2 + 4 + 6 + 8 + 10 + 12 = 42 Mean is given by, X = ∑x = 42 = 7 N 6

19. 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 = 117 Hence mean is given by X = ∑x = 117 = 23.4 N 6

20. 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.

21. Mean for grouped data Example 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, 4 Solution: Follow the below steps:

22. Mean for grouped data Example 1 Step 1: Prepare a frequency distribution table. ScoresTally marksFrequency 0II2 1I1 2 2 3III3 4IIIII7 5III3 6II2 N = 20

23. Mean for grouped data Example 1 Step 2: Calculate fx for each score by multiplying f and x. Add all fx to get ∑fx. Scores (x) Frequency (x) fx 020 111 224 339 4728 5315 6212 N = 20∑fx = 69

24. Mean for grouped data Example 1 Step 3: Now the mean is calculated as X = sum of the scores number of scores = ∑fx N = 69 20

25. Mean for grouped data Example 2 Find the mean of given frequency table. Class - IntervalFrequency 0 – 43 5 – 95 10 – 147 15 – 194 20 – 246 N = 25

26. Mean for grouped data Example 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.

27. Mean for grouped data Example 2 Class - IntervalMidpoint of Class Interval (x) Frequency (f) fx 0 – 4236 5 – 97535 10 – 1412784 15 – 1917468 20 – 24226132 N = 25∑fx = 325

28. Mean for grouped data Example 2 Step 4: Now the mean is calculated as X = sum of the scores total number of scores = ∑fx N = 325 = 13 25

29. Median It is the mid-point of the data after being arranged in asceding or descending order. Example: 8, 5, 3, 5, 6, 10, 7 Arranging in ascending order, 3, 5, 5, 6, 7, 8, 10 Hence median is 3 5 5 6 7 8 10

30. 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.

31. Median for an ungrouped data Example 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, 43 Here the number of terms is 7. The middle term is 4 th one and it is 37. Hence, median = 37

32. Median for an ungrouped data Example 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, 32 Here 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

33. 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, then Median is at (N + 1)/2 –th place If N is even, then Median is at ½(score at N/2 –th place + score at (N/2 + 1) –th place).

34. Median for grouped data Example Example 1: Find the median for the following grouped data. Class IntervalFrequency 1 – 54 6 – 103 11 – 156 16 – 205 21 – 252 N = 20

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

36. Median for grouped data Example Calculating cumulative frequency: Class IntervalFrequencyCumulative Frequency (f c ) 1 – 5444 6 – 10374 + 3 =7 11 – 156137 + 6 = 13 16 – 2051813 + 5 = 18 21 – 252208 + 2 =20 N = 20

37. Median for grouped data Example Solution: Step 3: Find median class. Count frequencies from first class interval and find the class interval that 10 th 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.

38. Median for grouped data Example Solution: So we have, LRL = 10.5 frequency of median class (f m ) = 6 cumulative frequency of median class (f c ) = 7 size of class interval (i) = 5

39. Median for grouped data Example Solution: Step 5: Calculate median using the formula median = LRL + ((N/2) – f c ) x i f m median = 10.5 + (20/2) – 7 6 = 13

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

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

42. 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, 5 Solution: Here 3 and 5 appear 4 times. Hence mode = 3 and 5

43. 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 Number1213141516 Frequency7962220

44. References https://www.khanacademy.org/math/cc-sixth- grade-math/cc-6th-data-statistics/cc-6th- statistics/v/mean-median-and-mode 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 https://www.youtube.com/watch?v=uhxtUt_- GyM