Tabulation of Data, Frequency Distribution, Contingency table

Frequency table

1. Tabulation Frequency Distributions Jagdish Powar Statistician cum Assistant Professor Communicate Medicine SMBT, IMSRC, Nashik.

2. Presentation of Data Principles of presentation of data: Increase interest of reader. Concise without losing important details. Presented in simple form. Facilitate further analysis. Define the problem & suggest it’s solution.

3. Methods of presentation of data Text Tabulation Diagrams & Graphs Tabulation is the first step in the analysis of data

4. In  statistics , a  frequency distribution  is a table that displays the frequency of various outcomes in a  sample . Each entry in the table contains the  frequency  or count of the occurrences of values within a particular group or interval, and in this way, the table summarizes the  distribution  of values in the sample. A  frequency distribution  shows us a summarized grouping of data divided into mutually exclusive classes and the number of occurrences in a class. It is a way of showing unorganized data. It is an organization of data into tables, graphs or charts, so that logical and statistical conclusions can be derived from the collected measurements. Frequency Distribution: Introduction

5. Frequency Distributions After collecting data, the first task for a researcher is to organize and simplify the data so that it is possible to get a general overview of the results. This is the goal of descriptive statistical techniques. One method for simplifying and organizing data is to construct a frequency distribution . 5

6. Frequency Distributions (cont.) A frequency distribution is an organized tabulation showing exactly how many individuals are located in each category on the scale of measurement. A frequency distribution presents an organized picture of the entire set of scores, and it shows where each individual is located relative to others in the distribution. 6

7. Frequency Distribution Tables A frequency distribution table consists of at least two columns - one listing categories on the scale of measurement (X) and another for frequency (f). In the X column, values are listed from the highest to lowest, without skipping any. For the frequency column, tallies are determined for each value (how often each X value occurs in the data set). These tallies are the frequencies for each X value. The sum of the frequencies should equal N. 7

8. Frequency Distribution The way, in which observations are distributed into various classes. Frequency distribution discrete frequency distribution continuous frequency distribution To make the work easy, we use tally marks.

9. Discrete Frequency Distribution :- When a frequency distribution table lists all of the individual categories (X values-i.e. Variables is Discrete) it is called a Discrete frequency distribution . 9

10. Ex:- II MBBS students conducted family health survey(FHS) and recorded number of children's among 40 families as below: Prepare the frequency table for given data & draw your conclusion from the same. 10 2 1 3 2 1 2 1 2 1 2 2 1 2 1 2 2 2 1 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 3 1 2 1

11. Soln : Let us consider the variable X : Number of children's in family S= Smallest value=Minimum value number of children’s =0, L= Largest Value= Maximum value number of children’s =3, 11 No. of children’s(X) Tally Marks Frequency (f) III 3 1 IIII IIII II 12 2 IIII IIII IIII IIII III 23 3 II 2 Total N= 40

12. Example:- No. of family members Tally bars frequency 1 IIII 5 2 IIII IIII 10 3 IIII IIII IIII 15 4 IIII IIII IIII IIII 20 5 IIII IIII 10 6 & more IIII 5 Total N= 65

13. Grouped (continuous) Frequency Distribution Sometimes, however, a set of scores covers a wide range of values. In these situations, a list of all the X values would be quite long - too long to be a “simple” presentation of the data. To remedy this situation, a grouped frequency distribution table is used. 13

14. Grouped Frequency Distribution (cont.) In a grouped table, the X column lists groups of scores, called class intervals , rather than individual values. These intervals all have the same width, usually a simple number such as 2, 5, 10, and so on. Interval must same throughout the all classes. Groups should not be too broad or too short. Group should be between 5 and 15. 14

15. Definition of Terms Range (R) – the difference between the highest score and the lowest score. Class Interval (k) – a grouping or category defined by a lower limit and an upper limit. Class Boundaries (CB ) – these are also known as the exact limits, and can be obtained by subtracting 0.5 from the lower limit of an interval and adding 0.5 to the upper limit interval.

16. Class Mark (x ) – is the middle value or the midpoint of a class interval. It is obtained by getting the average of the lower class limit and the upper class limit. Class Size ( i ) – is the difference between the upper class boundary and the lower class boundary of a class interval 7. Class Frequency – it refers to the number of observations belonging to a class interval, or the number of items within a category.

17. Example : Statistics Test Scores of 50 students. Construct a frequency distribution 51 65 68 87 76 56 69 75 89 80 61 66 73 86 79 70 71 54 87 78 68 74 66 88 77 67 73 64 90 77 72 52 67 86 79 74 59 70 89 85 55 63 74 82 84 57 68 72 81 83

18. Steps in Constructing a Frequency Distribution 1. Find the range R, using the formula : R = Highest Score – Lowest Score k 2. Compute for the number of class intervals, n, by using the formula: k = 1+3.3 log n

19. Note: The ideal number of class intervals should be 5 to 15. Less than 8 intervals are recommended for a data with less than 50 observations/values. For a data with 50 to 100 observations/values, the suggested number should be greater than 8.

20. 3. Compute for the class size, I, using the formula: i = R/k Please note also that the few number of class intervals will result to crowded data while too many number of class intervals tend to spread out the data too much. 4. Using the lowest score as lower limit, add (i – 1)to it to obtain the higher limit of the desired class interval.

21. 5. The lower limit of the second interval may be obtained by adding the class size to the lower limit of the first interval. Add (i – 1) to the result to obtain the higher limit of the second interval. 6. Repeat step 5 to obtain the third class interval, and so on, and so forth. 7. When the n class intervals are completed, determine the frequency for each class interval by counting the elements.

22. Solution : 1. R = Highest Score – Lowest Score R = 90 – 51 R = 39 2. k = 8 (desired interval ) 3. i = R/k i = 39/8 i = 4.875 i = 5

23. The Frequency Distribution of the Statistics Score of 50 Students Class Interval Tally Marks f LL - UL   50 - 55 IIII 4 55 - 60 III 3 60 - 65 IIII 4 65 - 70 IIII IIII 10 70 - 75 IIII IIII 9 75 - 80 IIII II 7 80 - 85 IIII 5 85 - 90 IIII III 8 Total N=50

24. Cumulative Frequency Distribution Class Interval f < cf >cf LL - UL       50 - 55 4 4 50 55 - 60 3 7 46 60 - 65 4 11 43 65 - 70 10 21 39 70 - 75 9 30 29 75 - 80 7 37 20 80 - 85 5 42 13 85 - 90 8 50 8 50 The Frequency Distribution of the Statistics Score of 50 Students

25. Cumulative frequency distribution: Cumulative frequency Less than cumulative frequency Greater than cumulative frequency ( l.c.f .) ( g.c.f .) l.c.f . denotes no. of observation whose values are less than upper limits. g.c.f . denotes no. of observation whose values are greater than lower limits.

26. Cumulative Frequency distribution Marks Frequency Cumulative frequency l.c.f . g.c.f . 0-50 5 5 60+5=N=65 50 - 60 10 5+10=15 50+10=60 60 - 70 15 15+15=30 35+15=50 70 – 80 20 30+20=50 15+20=35 80 - 90 10 50+10=60 10+5=15 90 – 100 5 N=65 5

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28. Contingency Table Or Two way Table for Qualitative data/ Categorical Data :-   A  two-way table  presents categorical data by counting the number of observations that fall into each group for two variables, one divided into rows and the other divided into columns .  

29. Contingency Table Or Table for Qualitative data/ Categorical Data :- There are 40 students in batch D of II MBBS out of which 25 are boys. The 18 boys and 13 girls lives in hostel. Prepare contingency table for gender wise current residential status for batch D.

30. Put the given values in table Sex\Residential Status Hostelite Day Scholar Total Male 18 ?? 25 Female 13 ?? ?? Total ?? ?? N= 40

31. Ex. In study of susceptibility to diphtheria, 982 children were studied in Mumbai. Out of 494 boys 63 were Schick positive i.e. susceptible. 807 children were schick negative and out of these 376 were girls. Prepare table for Given Information

32. Classification of children according to sex and susceptibility to diphtheria Schick Test Sex + ve - ve Total Boys 63 ?? 494 Girls ?? 376 ?? Total ?? 807 N= 982

33. Schick Test Sex + ve - ve Total Boys 63 =494 -63 =431 494 Girls =492-376 =112 376 =982-492 =488 Total =63+112 =175 807 N= 982

34. Solution Ex1. L= largest Value=10.1 S= Smallest value=13.9 R=L-S=13.9-10.1=3.8 k=1+1.3322* log(N)=1+3.322*2 ≈8 i=R/k=3.8/8=0.475≈0.5 34

35. Thank You 35