CHAPTER 3 : DESCRIPTIVE STATISTIC : NUMERICAL MEASURES (STATISTICS)

DESCRIPTIVE STATISTICS : NUMERICAL MEASURES (STATISTICS) 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,

2002 Total Payroll (Million of dollars) Example 3.1: The mean sample of CGPA (raw/ungroup) is: Table 3.1 MLB Team 2002 Total Payroll (Million of dollars) Anaheim Angels 62 Atlanta Braves 93 New York Yankees 126 St. Louis Cardinals 75 Tampa Bay Devil Rays 34 Total 390

- For tabular/group data, mean is defined by:

Example 3.2 : The mean sample for Table 3.2 CGPA (Class) Frequency, f Class Mark (Midpoint), x fx 2.50 - 2.75 2 2.625 5.250 2.75 - 3.00 10 2.875 28.750 3.00 - 3.25 15 3.125 46.875 3.25 - 3.50 13 3.375 43.875 3.50 - 3.75 7 3.625 25.375 3.75 - 4.00 3 3.875 11.625 Total 50   161.750 Table 3.2

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.

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.

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 - 2.75 2 2.75 - 3.00 10 12 3.00 - 3.25 15 27 3.25 - 3.50 13 40 3.50 - 3.75 7 47 3.75 - 4.00 3 50 Total   Table 3.3

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.  

*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.  

- Determining the mode using formula. *Note: - The class which has the highest frequency is called the modal class.

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 - 2.75 2 2.75 - 3.00 10 3.00 - 3.25 15 3.25 - 3.50 13 3.50 - 3.75 7 3.75 - 4.00 3 Total 50 Modal Class Table 3.4

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 s2 and the sample standard deviation is denoted by s.

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. State Total Area (square miles) Arkansas 53,182 Louisiana 49,651 Oklahoma 69,903 Texas 267, 277 Table 3.4

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 s2 and defined by: ii) Variance for grouped data The variance for the frequency distribution is defined by:

Example: Ungrouped Data 7 , 6, 8, 5 , 9 ,4, 7 , 7 , 6, 6 Range = 9-4=5 Mean Variance Standard Deviation

Example: Ungrouped Data 7 , 6, 8, 5 , 9 ,4, 7 , 7 , 6, 6

Example 3.9 for grouped data : The variance for frequency distribution in Table 3.5 is: CGPA (Class) Frequency, f Class Mark, x fx fx2 2.50 - 2.75 2 2.625 5.250 13.781 2.75 - 3.00 10 2.875 28.750 82.656 3.00 - 3.25 15 3.125 46.875 146.484 3.25 - 3.50 13 3.375 43.875 148.078 3.50 - 3.75 7 3.625 25.375 91.984 3.75 - 4.00 3 3.875 11.625 45.047 Total 50   161.750 528.031 Table 3.5

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

Example 3. 10 (Based on example 3. 8) for ungrouped data: Example 3.10 (Based on example 3.8) for ungrouped data: *Refer example Example 3.11 (Based on example 3.9) for grouped data:

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: Determine the interval Find value of Change the value in step 2 to a percent Write statement: at least the percent of data found in step 3 is in the interval found in step 1

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: Determine interval Find Convert into percentage: Conclusion: At least 75% of the data is found in the interval from 70 to 90

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

Example The age distribution of a sample of 5000 persons is bell-shaped with a mean of 40 yrs and a standard deviation of 12 yrs. Determine the approximate percentage of people who are 16 to 64 yrs old. Solution: 95% of the people in the sample are 16 to 64 yrs old.

Exercise for summarizing data The following data give the total number of iPods sold by a mail order company on each of 30 days. Construct a frequency table. Find the mean, variance and standard deviation, mode and median. 23 14 19 23 20 16 27 9 21 14 22 13 26 16 18 12 9 26 20 16 8 25 11 15 28 22 10 5 17 21 Institut Matematik Kejuruteraan, UniMAP