# Measures of Central Tendency, Box Plots, and Outliers

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Q1.Measures of Central TendencyThe measures of central tendency is a parameter used to specify the point where the dataare centered. This method of measures of central tendency is widely used than any otherstatistical measure because they are computed easily and can be applied easily. Hence aset of data is described in this by identifying the central position of data. The common measures of central tendency are like arithmetic mean, the median, themode, the weighted mean, and the geometric mean. Such measures of central tendencyare all applied under some conditions which are given below.1) Arithmetic Mean: This computes the sum of all the observations divided by the numberof observations. This is used when there is a need to measure a middle or centre of dataand when the type of variable is interval/ratio (not skewed).2) The median: This computes the middle value in a given set of items that are sorted inascending or descending order. Such a measure of central tendency is used when arelative position of the given observations is to be computed. It is generally preferredwhen the data is skewed and ordinal.3) The mode: In a given data set the value which occurs most frequently is known as themode. This type of measure of central tendency is generally used with nominal data.Q2.Stem Plot is a table which is special since the first digits of the number are split into astem and last digits are structured into a leaf. This is a method of representing thefrequency and presenting the quantitative data in a graphical format.Now in this case first rearrange the data from smallest data to largest data. So it becomesas 55 56 59 62 63 64 64 64 69 77 77 85Now take the first digit of the smallest number 55 and so onHence the stem plot for given data setStemLeaves55 6 962 3 4 4 4 97 7 78 5Mean of the given data is (55 + 77 + 64 + 77 + 69 + 63 + 62 + 64 + 85 + 64 + 56 + 59)/1266.25Q3.(a)Mean = sum of all observations / total no. of observations(27 + 30 + 21 + 62 + 28 + 18 + 23 + 22 + 26 + 28) / 1028.5hence mean = 28.5As number of observations are even in number thus the median is the mean of the valuesof observations occupying n/2 and (n+2)/2 Hence n/2 = 10/2 = 5th position(n+2)/2 = 12/2 = 6th positionfirst sort the observations in ascending order which comes as 18 21 22 23 26 27 28 30 62thus at 5th position value is 26at 6th position value is 27mean of the numbers at 5th and 6th position is = (26 + 27)/226.5 hence median = 26.5(b)In the above scenario mean = 28.5 and median = 26.5, therefore median is the bestmeasure of central tendency as the distribution above is skewed and have the smallnumber of observations.(c)Outliers is defined as the point of observation that is too far or at a distant from otherobservations. Thus, the outlier in above given set of observations is the value 62.Q4.Four test scores are 74%, 68%, 84%, and 79%We have to find his fifth test score in exam and his average marks given are 75%Let x be his fifth test score thus (74 + 68 + 84 + 79 + x)/5 = 75305 + x = 375x = 375 – 305x = 70Hence, the minimum score he needs on the final exam to pass the class with 75%average is equal to 70%

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