1. CHAPTER 3 : DESCRIPTIVE STATISTIC : NUMERICAL MEASURES (STATISTICS)

2. 3.1 Measures of Central Tendency/ Location There are 3 popular central tendency measures, mean, median & mode. 1) Mean  The mean of a sample is the sum of the measurements divided by the number of measurements in the set. Mean is denoted by ( ) Mean = Sum of all values / Number of values  Mean can be obtained as below :- - For raw data, mean is defined by,

3. - For tabular/group data, mean is defined by: Where f = class frequency; x = class mark (mid point) Example 3.1: The mean sample of CGPA (raw/ungroup) is: MLB Team 2002 Total Payroll (Million of dollars) Anaheim Angels 62 Atlanta Braves93 New York Yankees126 St. Louis Cardinals75 Tampa Bay Devil Rays34 Total390 Table 3.1

4. Example 3.2 : The mean sample for Table 3.2 CGPA (Class)Frequency, f Class Mark (Midpoint ), xfx 2.50 - 2.7522.6255.250 2.75 - 3.00102.87528.750 3.00 - 3.25153.12546.875 3.25 - 3.50133.37543.875 3.50 - 3.7573.62525.375 3.75 - 4.0033.87511.625 Total50 161.750 Table 3.2

5. 2) Median  Median is the middle value of a set of observations arranged in order of magnitude and normally is devoted by i) The median for ungrouped data. - The median depends on the number of observations in the data,. - If is odd, then the median is the th observation of the ordered observations. - If is even, then the median is the arithmetic mean of the th observation and the th observation.

6. ii) The median of grouped data / frequency of distribution. The median of frequency distribution is defined by: where, = the lower class boundary of the median class; = the size of the median class interval; = the sum of frequencies of all classes lower than the median class = the frequency of the median class.

7. Example 3.3 for ungrouped data :- The median of this data 4, 6, 3, 1, 2, 5, 7, 3 is 3.5. Proof :- - Rearrange the data in order of magnitude becomes 1,2,3,3,4,5,6,7. As n=8 (even), the median is the mean of the 4th and 5th observations that is 3.5. Example 3.4 for grouped data :- CGPA (Class)Frequency, f Cum. frequency 2.50 - 2.7522 2.75 - 3.001012 3.00 - 3.2515 27 3.25 - 3.501340 3.50 - 3.75747 3.75 - 4.00350 Total50 Table 3.3

8. 3) Mode The mode of a set of observations is the observation with the highest frequency and is usually denoted by ( ). Sometimes mode can also be used to describe the qualitative data. i) Mode of ungrouped data :- - Defined as the value which occurs most frequent. - The mode has the advantage in that it is easy to calculate and eliminates the effect of extreme values. - However, the mode may not exist and even if it does exit, it may not be unique.

9. *Note:  If a set of data has 2 measurements with higher frequency, therefore the measurements are assumed as data mode and known as bimodal data.  If a set of data has more than 2 measurements with higher frequency so the data can be assumed as no mode. ii) The mode for grouped data/frequency distribution data. - When data has been grouped in classes and a frequency curve is drawn to fit the data, the mode is the value of corresponding to the maximum point on the curve.

10. - Determining the mode using formula. where = the lower class boundary of the modal class; = the size of the modal class interval; = the difference between the modal class frequency and the class before it; and = the difference between the modal class frequency and the class after it. *Note: - The class which has the highest frequency is called the modal class.

11. Example 3.5 for ungrouped data : The mode for the observations 4,6,3,1,2,5,7,3 is 3. Example 3.6 for grouped data based on table : Proof :- CGPA (Class)Frequency 2.50 - 2.752 2.75 - 3.00 10 3.00 - 3.2515 3.25 - 3.5013 3.50 - 3.757 3.75 - 4.003 Total50 Table 3.4 Modal Class

12. 3.2 Measure of Dispersion  The measure of dispersion/spread is the degree to which a set of data tends to spread around the average value.  It shows whether data will set is focused around the mean or scattered.  The common measures of dispersion are: 1) range 2) variance 3) standard deviation  The standard deviation actually is the square root of the variance.  The sample variance is denoted by s 2 and the sample standard deviation is denoted by s.

13. 1) Range  The range is the simplest measure of dispersion to calculate. Range = Largest value – Smallest value Example 3.7:- Table 3.5 gives the total areas in square miles of the four western South- Central states the United States. Solution: Range = Largest Value – Smallest Value = 267, 277 – 49, 651 = 217, 626 square miles. StateTotal Area (square miles) Arkansas53,182 Louisiana49,651 Oklahoma69,903 Texas267, 277 Table 3.4

14. 2) Variance i) Variance for ungrouped data  The variance of a sample (also known as mean square) for the raw (ungrouped) data is denoted by s 2 and defined by: ii) Variance for grouped data  The variance for the frequency distribution is defined by:

15. Example 3.8 for ungrouped data : Refer example. Example 3.9 for grouped data : The variance for frequency distribution in Table 3.5 is: CGPA (Class)Frequency, f Class Mark, xfxfx 2 2.50 - 2.7522.6255.25013.781 2.75 - 3.00102.87528.75082.656 3.00 - 3.25153.12546.875146.484 3.25 - 3.50133.37543.875148.078 3.50 - 3.7573.62525.37591.984 3.75 - 4.0033.87511.62545.047 Total50 161.750528.031 Table 3.5

16. 2Frequency, f Class Mark, xfxfx 2 2.50 - 2.7522.6255.25013.781 2.75 - 3.00102.87528.75082.656 3.00 - 3.25153.12546.875146.484 3.25 - 3.50133.37543.875148.078 3.50 - 3.7573.62525.37591.984 3.75 - 4.0033.87511.62545.047 Total50 161.750528.031

17. 3) Standard Deviation i) Standard deviation for ungrouped data :- ii) Standard deviation for grouped data :-

18. Example 3.10 (Based on example 3.8) for ungrouped data: *Refer example Example 3.11 (Based on example 3.9) for grouped data:

19. 3.3 Rules of Data Dispersion By using the mean and standard deviation, we can find the percentage of total observations that fall within the given interval about the mean. i) Chebyshev’s Theorem At least of the observations will be in the range of k standard deviation from mean. where k is the positive number exceed 1 or (k>1). Applicable for any distribution /not normal distribution. Steps: 1) Determine the interval 2) Find value of 3) Change the value in step 2 to a percent 4) Write statement: at least the percent of data found in step 3 is in the interval found in step 1

20. Example 3.12 : Consider a distribution of test scores that are badly skewed to the right, with a sample mean of 80 and a sample standard deviation of 5. If k=2, what is the percentage of the data fall in the interval from mean? Solution: 1) Determine interval 2) Find 3) Convert into percentage: 4) Conclusion: At least 75% of the data is found in the interval from 70 to 90

21. ii) Empirical Rule Applicable for a symmetric bell shaped distribution / normal distribution. k is a constant. k is a 1, 2 or 3 for Empirical Rule. There are 3 rules: i. 68% of the observations lie in the interval ii. 95% of the observations lie in the interval iii. 99.7% of the observations lie in the interval If k is not given, then: Formula for k =Distance between mean and each point standard deviation

23. ClassFrequency (f)Midpoint (x m )f. x m 5.5-10.5188 10.5-15.521326 15.5-20.531854 20.5-25.5523115 25.5-30.5428112 30.5-35.533399 35.5-40.523876 1 n=20

24. Arrange the data in order 209, 211, 211, 212, 213, 223, 227, 229, 240, 240 median

25. 3. The median class is 5-6, since it contains the 5th value, (n/2 =5. From the table, Movies Showing Frequenc y, f Cummulative frequency Class Mark, xx2x2 fx fx 2 1-21 11.52.251.52.25 3-42 33.512.25724.5 5-63 65.530.2516.590.75 7-81 77.556.257.556.25 9-103 109.590.2528.5270.75