Measures of centerComparison of the Mean, Median, and ModeRelative positions of mean and medianMeasures of variationFrequency HistogramConsider the problem of analyzing the distribution of test scores (see table below) in acertain test given to 35 students.Raw Test Scores (in %)73 75 67 76 6272 86 68 79 7581 76 69 83 7385 77 71 84 8293 83 73 86 8765 84 75 77 9267 92 74 78 94We call the collection of all 35 test scores the population and the test scores given just asthey are in the table without any sort of arrangement, the raw data. Let us assume thegrading scheme to be the usualA 90% - 100%B 80% - 89.99%C 70% - 79.99%D 60% - 69.99%E < 60%To begin the analysis, we would be interested in knowing the grade distribution for theclass. We do this in the following steps:I. Arrange the raw data in ascending order to form an array.6265676768697172737373747575757676777778798182838384848586868792929394II. The range of the data is defined to be the difference between the largest and the smallest numbers in the array: Range of data above = 94 – 62 = 32III. The width of a class-interval is obtained by dividing the range of the data by the number of class intervals desired. Since we are interested in the grade distribution we choose the width to be 10, and starting with the score of 60% we use the class-intervals 60 – 69, 70 – 79, 80 – 89, 90 – 100. IV. We then determine the class frequencies, which are the number of scores that lie in each class-interval. The relative frequency of a class-interval is the frequency of the class-interval divided by the total number of scores. The class frequencies and relative frequencies for our data are shown in the table below:Class-Intervals Class Frequencies Relative Frequencies60 - 69 6 6/3570 - 79 15 15/3580 - 89 10 10/3590 - 100 4 4/35Total 35 1A visual of the frequency table are shown in the bar graphs below:Frequency HistogramRelative Frequency HistogramExample: Consider the data given the table. The data is sorted in ascending order in thesecond column. DATA SORTED DATA CLASSES TALLY FREQUENCY130 45 40 - 49 1 158 50 50 - 59 11111 11 797 51 60 - 69 11111 11 754 54 70 - 79 111 396 55 80 - 89 11111 1 655 55 90 - 99 11111 11111 10101 55 100 - 109 11111 577 58 110 - 119 1111 486 60 120 - 129 11 287 62 130 - 139 111 345 64 140 - 149 075 66 150 - 159 11 251 66 TOTAL 50100 67129 6964 75111 7767 7878 80109 81155 83151 83125 8693 8769 9066 91139 9350 94113 9694 9660 9781 97136 97111 9999 10080 10155 10255 104104 10997 111102 11166 11383 11396 12583 12962 13090 13691 139113 15197 155Mean 89.72Median 90.5Mode 97Measures of centerMeasures that describe where the center or most typical value of a data set lies are called measures of central tendency. The three most important measures of center are mean, median, and mode.Mean: The mean of a data set of values is the sum of the data values divided by the total number of data values.Median: To find the median, we first arrange the data in ascending order. - If the number of data values is odd, then the median is exactly the middle value of theordered list.- If the number of data values is odd, the median is the average of the two middle data values.Mode: If the frequency of occurrence of a data value is greater than or equal to two, then the value that occurs with the greatest frequency is called the mode of the data set. If no value of the data set occurs more than once, the data set has no mode.Comparison of the Mean, Median, and ModeGenerally, the mean is sensitive to extreme observations, whereas the median is not. Consequently the median is usually preferred as the measure of center for date sets that have extreme observations.Relative positions of mean and median1. Right-skewed Median Mean2. Symmetric Median Mean3. Left-skewed Mean MedianNote: For right-skewed: the mean is greater than median For symmetric: the mean and median are equal For left-skewed: the mean is less than the medianMeasures of variationMeasures that indicate the amount of variation, or spread, in a data set, are called measures of variation. One measure of variation is the standard deviation which measure how much, on average, the data values vary from the mean. The standard deviation is calculated in four steps:- We first compute the deviation of each data value from the mean.- We then calculate the sum of squared deviations - We then calculate the sample variance by dividing the sum of the squared deviations by n – 1 if the sample size is n.- The sample standard deviation is the square of the sample variance. The steps above give the following formula for sample standard deviation:1)(2nxxsXo Assignment:- Question: Are men taller than women? Think about how would you answer this for next class- Each student should find the height of friends (5 men and 5 women) and email them to (M) no later than MondayClass 2- Question: Are men taller than women?- Sampling – pick one man and one woman from class and compare (if possible, purposefully pick a short man and tall woman)- Was this sampling random?- Pick another pair (if possible, pick a tall man and short woman) as a contrast.- Would our conclusions be different if we chose to sample from a basketball court?A gymnastics exhibition?- Importance of sample size.- Compare heights of class to heights from their friends- What do the distributions look like.- Plot data, calculate averages, etc.- Are students in the class taller or shorter than their friends?- Male versus