Mean for population data: x = the sum of all values. N = the population size n = the sample size, µ = the population mean. x = the sample mean

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1 MEASURE OF CENTRAL TENDENCY MEASURS OF CENTRAL TENDENCY Ungrouped Data Measurement Mean Mean for population data: Mean for sample data: x N x x n where: x = the sum of all values N = the population size n = the sample size, µ = the population mean x = the sample mean The following data give the prices (rounded to thousand $) of five homes sold recently in Sekayang Find the mean sale price for these homes. Thus, these five homes were sold for an average price of 186 thousand The mean has the advantage that its calculation includes each value of the data set. Weighted Mean Used when have different needs. Weight mean : MEASURE OF CENTRAL TENDENCY 1

2 x w wx w where w is a weight. Consider the data of electricity components purchasing from a factory in the table below: Type Number of component (w) Cost/unit (x) Total 6000 RM3.00 RM3.40 RM.80 RM.90 RM3.5 x w wx w 100(3) 500(3.4) 500(.8) 1000(.9) 800(3.5) = = 6000 =.967 Mean cost of a unit of the component is RM.97 Median Median is the value of the middle term in a data set that has been ranked in increasing order. Procedure for finding the Median Step 1: Rank the data set in increasing order. Step : Determine the depth (position or location) of the median. Depth of Median = n 1 MEASURE OF CENTRAL TENDENCY

3 Step 3: Determine the value of the Median. Find the median for the following data: (1) Rank the data in increasing order () Determine the depth of the Median n 1 Depth of Median = 51 = = 3 (3) Determine the value of the median Therefore the median is located in third position of the data set. Hence, the Median for above data = Find the median for the following data: (1) Rank the data in increasing order () Determine the depth of the Median n 1 Depth of Median = 61 = = 3.5 (3) Determine the value of the Median Therefore the median is located in the middle of 3 rd position and 4 th position of the data set. MEASURE OF CENTRAL TENDENCY 3

4 8 10 Median 9 Hence, the Median for the above data = The median gives the center of a histogram, with half of the data values to the left of (or, less than) the median and half to the right of (or, more than) the median. The advantage of using the median is that it is not influenced by outliers. Mode Mode is the value that occurs with the highest frequency in a data set. 1. What is the mode for given data? What is the mode for given data? Mode =. Mode = A major shortcoming of the mode is that a data set may have none or may have more than one mode. One advantage of the mode is that it can be calculated for both kinds of data, quantitative and qualitative. Grouped Data Measurement Mean Mean for population data: μ= N fx Mean for sample data: MEASURE OF CENTRAL TENDENCY 4

5 fx x= n Where x the midpoint and f is the frequency of a class. The following table gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company. Calculate the mean. Number of order f n = 50 Because the data set includes only 50 days, it represents a sample. The value of calculated in the following table: Number of order f x fx n = 50 fx is The value of mean sample is: Thus, this mail-order company received an average of orders per day during these 50 days. Median Step 1: Construct the cumulative frequency distribution. Step : Decide the class that contain the median. Class Median is the first class with the value of cumulative frequency is at least n/. Step 3: Find the median by using the following formula: n -F Median = L m + i fm Where: n = the total frequency F = the total frequency before class median i = the class width L m = the lower boundary of the class median f m = the frequency of the class median MEASURE OF CENTRAL TENDENCY 5

6 Based on the grouped data below, find the median: Time to travel to work Frequency st Step: Construct the cumulative frequency distribution Time to travel to work Frequency Cumulative Frequency Thus, 5 persons take less than 4 minutes to travel to work and another 5 persons take more than 4 minutes to travel to work. Mode Mode is the value that has the highest frequency in a data set. For grouped data, class mode (or, modal class) is the class with the highest frequency. Formula of mode for grouped data: Δ 1 Mode = L mo + i Δ 1+ Δ Where: L mo 1 i is the lower boundary of class mode is the difference between the frequency of class mode and the frequency of the class before the class mode is the difference between the frequency of class mode and the frequency of the class after the class mode is the class width MEASURE OF CENTRAL TENDENCY 6

7 Based on the grouped data below, find the mode Time to travel to work Frequency Based on the table, We can also obtain the mode by using the histogram; Relationship among Mean, Median & Mode As discussed in previous topic, histogram or a frequency distribution curve can assume either skewed shape or symmetrical shape. Knowing the value of mean, median and mode can give us some idea about the shape of frequency curve. (1) For a symmetrical histogram and frequency curve with one peak, the value of the mean, median and mode are identical and they lie at the center of the distribution. MEASURE OF CENTRAL TENDENCY 7

8 Mean, median, and mode for a symmetric histogram and frequency distribution curve () For a histogram and a frequency curve skewed to the right, the value of the mean is the largest that of the mode is the smallest and the value of the median lies between these two. Mean, median, and mode for a histogram and frequency distribution curve skewed to the right (3) For a histogram and a frequency curve skewed to the left, the value of the mean is the smallest and that of the mode is the largest and the value of the median lies between these two. Mean, median, and mode for a histogram and frequency distribution curve skewed to the left MEASURE OF CENTRAL TENDENCY 8

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