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The Mode● The mode is defined to be the value that occurs most often in a data set ● A data set can have more than one mode (bimodal, unimodal, trimodal) ● A data set is said to have no mode if all values occur with equal frequency ● Ex. Data Set: 1,1,2,4,2,5,2Ordered Set: 1,1,2,2,2,4,5 ○ Mode= 2 ● No Mode ○ Data Set= 1,2,3,4,5,6 No mode ● Two modes ○ Data Set= 1,1,2,2,3,4,5,6 Mode=1,2 ● The mode for a grouped frequency distribution ● Class F 15.520.5 3 20.525.5 5 25.530.5 7 30.535.5 3 35.540.5 2 ○ Modal Class= 25.530.5 ● The mode for an ungrouped frequency distribution ○ Values F 15 3 20 5 25 8 30 3 35 2 ■ The mode for grouped data is the modal class ■ The modal class is the class with the largest frequency ■ Sometimes the midpoint of the class is used rather than the boundaries ■ Mode=8 ● When to use the mode? ○ When the variable being examined is nominal, only the mode can be used as a measure of central tendency ○ When you want to describe the most common value of the distribution
The Median ● When a data set is ordered, called data array ● The median is defined to be the midpoint of the data array ● The symbol for median= MD ● There are as many values of the variable below the median as above the median value ○ Ex. 190, 230, 195, 210, 245 ○ Data Array: 190, 195, 210, 230, 245MD= 210 ● In an odd number of values in the data set, it is easy to select the middle number in the data array ● In an even number of values in the data set, the median is obtained by taking the avg. of the two middle numbers ○ Ex. 10, 20, 30, 40, 50, 60 ○ Data Array: 10, 20, 30, 40, 50, 60 MD=(30+40)/2= 35 ● The median: Ungrouped Frequency Distribution ○ For an ungrouped frequency distribution, find the median by examining the cumulative frequencies to locate the middle value ○ If ‘n’ is the sample size, compute n/2. Locate the data point where n/2 values fall below and n/2 values fall above ○ Ex. Class Frequency 1 4 2 9 3 6 4 3 5 2 ■ To locate the middle point divide 24/2= 12 ■ Locate the point where 12 values would fall below and 12 values will fall above ■ Consider the cumulative distribution ■ The 12th and 13th value fall in class 2 MD=2 ● When to use the Median? ○ When the variable of interest is ordinal, internal, or ratio ○ When the mean would a misleading value of the typical case The Mean ● The mean is defined to be the sum of the data values divided by the total number of values ● Think of the mean as the equilibrium of a scale. It balance the data set.
● We will compute two means and one for the sample and one for a finite population of values ● The mean is not an actual data value ● The sample mean ○○ Ex. 1,3,5,7,9 ■ Sample mean= (1+3+5+7+9)/5= 25/5= 5 ● Population Mean ○ N= Size of the populationMu= Population Mean ○ Mu= (20,000+40,000+60,000+80,000)/4= 200,000/4= 50,000 ● What does the word “average” mean? ○ It is ambiguous, since several different methods can be used to obtain an average ○ The average means the center of the distribution, or the most typical case in the data ○ Measures of average are also called measure of central tendency, and include the mode, median, mean, and midrange ● The Weighted Mean ○ A set of number X1,X2,...,Xn with corresponding weights W1,W2,...,Wn is computed from the following formula
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School: Indiana University
Course: Statistical Techniques
Professor: Noah Hammarlund
Term: Spring 2018
Name: SPEA-K 300 Ch.3 Notes
Description: Central Tendency and Measures of Variances