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Measures of Central Tendency Statistics For Business MATH0201 The arithmetic mean is the best known type of average and is widely understood. It is used for further statistical analysis. “x bar” Advantages • widely understood •It is easy to calculate •Represents whole set of data •Suited to further statistical analysis Disadvantages •Value may not correspond to an actual value •Distorted by extreme values Eg. Mean : ungrouped data Find arithmetic mean of data in a frequency distribution = ∑fx / ∑f x = 185 / 20 = 9.25 Find arithmetic mean of grouped data • Find mid-point of each class interval. Arithmetic mean x = ∑ fx / ∑f = 190 / 20 = 9.5 units The median is the value of the middle item of a distribution once all of the items have been arranged in order of magnitude. • The median of the following nine values: 8 6 9 12 15 6 3 20 11 • The median of the following ten values: 8 6 9 12 15 6 3 20 11 7 The median of ungrouped frequency distribution remember to arrange the items in order of value first The median of a grouped frequency distribution can be estimated from an ogive The mode is an average which is the most frequently occurring value. Advantages • easy to find •Uninfluenced by a few extreme values •Can be used for nonnumerical data Disadvantages •Does not take every value into account •Some instability in the measure The mode of a grouped frequency distribution can be estimated from a histogram Modal class of a histogram is always the class with the tallest bar. Graphical method: using formula Let L be the lower class boundary of the model class U be the upper class boundary of the modal class Hm be the height of the tallest histogram block Hm- be the height of the block before the tallest one Hm+be the height of the block following the tallest one Then the mode is given by the formula: Mode = L + Hm - Hm(Hm - Hm-) + (Hm - Hm+) X (U – L) Eg. Invoice value (US$) Frequency, f 2 and under 6 6 and under 10 10 and under 14 14 and under 18 18 and under 22 22 and under 30 1 6 6 10 8 9 Frequency density 0.250 1.500 1.500 2.500 2.000 1.125