1. Statistical Inference for Managers Lecture-2 By Imran Khan

2. Class limits and class boundaries Class limits/ class boundaries: If there is some gap between the intervals then points defining the intervals are called class limits. Example: 0-4, 5-9, 10-14, 15-19. When the intervals are overlapping such that end point of one interval is same as the starting point of next interval, such intervals are called class boundaries. We use class boundaries for mean, median, Q1, Q3, Histogram and cumulative frequency curve. To convert class limits into class boundaries, we find the gap between end point of one class and starting point of next class and half a gap is subtracted from lower limits and half a gap is added to upper limits.

3. Class limits and class boundaries Class limits Class boundaries 2-4 1-5 6-8 5-9 10-12 9-13 14-16 13-19 Gap=6-4=2 Half= 2/2= 1

4. Histogram Histogram: It is a set of adjacent rectangles whose width is equal to the width of intervals or height is proportional to the frequency of intervals (in case of equal width intervals).  In a histogram graph, we plot the Frequency along y-axis against class boundaries along x-axis. Incase of un-equal width intervals, we calculate frequency density. Frequency density= frequency/ class width- In this case height is proportional to the frequency density.

5. Frequency Density Example-1: Mark frequency width FD 0-40 10 40 0.25 40-60 20 20 1 60-70 50 10 5 70-100 15 30 0.5 Plot graph?

6. Histogram Example-2: f Class boundaries 10-14 3 9.5-14.5 15-19 5 14.5-19.5 20-24 10 19.5-24.5 25-29 2 24.5-29.5 Histogram???

7. Mode Calculation of mode from Histogram: f 5-10 2 10-15 3 15-20 5 20-25 2

8. Histogram questions (continued) Frequency curve/ Frequency polygon: If we plot the points taking frequency along y-axis and mid-points of the intervals along x-axis. If we join the points by a straight line segment then it is called frequency polygon.

9. Cumulative Frequency Cumulative Frequency curve: Points are plotted taking cumulative frequency along Y-axis against upper class boundaries along X-axis. Quartiles and median of a grouped data can be calculated through cumulative frequency curve.

10. Cumulative Frequency curve f CF 0-10 3 10-20 4 20-30 7 30-40 16 40-50 5 50-60 2 For Q1, we find point on Y-axis corresponding to Σf/4 then join that point to CF curve. Draw CF curve? Q1= 37/4=9.28 th =23 Q2=37/2=18.5 th =33 Q3=3(37)/4=27.75 th =39

11. Stem and Leaf Diagram Plot a stem and leaf diagram? 23, 25, 43, 20, 14, 18, 07 45, 32, 82, 75, 63, 64, 35, 37, 38 40, 30, 35, 40, 42, 87, 93 97, 11, 6, 5, 12, 95, 49, 14 Key: 9/7 represents 97

12. Stem and leaf diagram 219, 541, 58, 7, 13, 476, 418, 177, 175, 455, 258, 312, 164, 314, 336, 121, 77, 183, 133, 78, 291, 138, 244, 36, 48 Plot stem and leaf diagram for the data? Make a box and viscar diagram? Comment about the skewness of the data?

13. Histogram-Tally method! Example: 76, 59, 93, 87, 38, 50, 56, 113 45, 67, 102, 34, 54, 85, 85, 50 44, 33, 51, 40, 82, 92, 79, 38 86, 34, 29, 107, 63, 46, 68, 49 99, 62, 61,72,79, 48, 70, 40 Draw Histogram? Minimum value of the data= 29 Maximum value of the data= 113 Range= 113-29= 84 Class width= range/no of classes= 84/10= 8.4

14. Histogram-Tally method! Plot a histogram? Steps:  Determine no of classes  Find Class width  Create classes  Histogram

15. End of lecture! Thank you!