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Descriptive Measures “While an individual is an insolvable puzzle, in an aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man do, but you can say with precision what an average number will be upto”. -Arthur Conan 1 Measures of Central Tendency Mean Median Mode 2 Requisites of a good measure of central tendency 1. 2. 3. 4. 5. It should be easy to calculate and understand. It should be rigidly defined. It should be representative of the data. It should have sampling stability. It should not be affected by extreme values. 3 Which measure is appropriate? A person who does not know how to swim has to cross a river from one bank to another by walking into the water with a stick. He tries to determine the length of stick on the basis of average length of river. Which measure of central location w.r.t the depth of the river will save him from getting drowned? 4 Mean 5 The FD below represents time in seconds needed to serve a sample of customers by a cashier at a discount store. Find the Mean. 6 Time 20-30 30-40 40-50 50-60 60-70 70-80 Freq 6 16 21 29 25 22 Time 80-90 90-100 100-110 110-120 120-130 Freq 11 7 4 0 2 7 Properties of Mean 1. 2. 3. The sum of deviations of all individual observations from mean is always zero. The sum of squares of deviations of observations about the mean is minimum. Combine mean of set groups can be obtained. 8 X X-Mean Sqr (X-Mean) Sqr (X-10) 10 -20 400 0 20 -10 100 100 30 0 0 400 40 10 100 900 50 20 400 1600 Sum=1000 Sum=3000 Mean=30 Sum=0 9 Limitations 1. 2. 3. 4. Average may give a value that does not exist in data Average may give absurd results Does not give idea about difference in series Affected by extreme values 10 Weighted Arithmetic Mean 11 Dave’s Giveaway Store advertises, “If our prices are not equal or lower than everyone else’s, you get it free.” One of Dave’s customers came into the store one day and threw on the counter bills of sale for six items she bought from a competitor for an average price less than Dave’s. 12 The items cost : $1.29, $2.97, $3.49, $5.00, $7.50,$10.95. Dave’s prices for the same six items are $1.35, $2.89, $3.19, $4.98, $7.59, $11.50 Dave told, “My add refers to a weighted average price for the same items. Our average is low because our sales of these items have been: 7, 9, 12, 8, 6, 3. 13 Is Dave getting himself into our out of trouble by talking about weighted average? 14 Median 15 Ungrouped distribution 2 5 10 20 30 33 57 16 3 10 13 17 33 60 17 Meridian Trucking company maintains records on all its rolling equipment. Here are weekly mileage records for its trucks. Calculate the median miles a truck travelled. 810 450 756 789 210 657 589 488 878 689 1450 560 469 890 987 559 788 943 447 775 18 Median = 722.5 Miles 19 Grouped distribution Class Freq Class Freq 18-22 120 38-42 184 22-26 125 42-46 162 26-30 280 46-50 86 30-34 260 50-54 75 34-38 155 54-58 53 20 Median = 33.46 21 Calculate median from the data pertaining to profits (in Cr.) of 125 companies: Profits Less Less Less Less than than than than 10 20 30 40 No. of comp 4 16 40 76 Profits Less Less Less Less than than than than 50 60 70 80 No. of comp 96 112 120 125 22 Median Profit = 36.25 Cr. 23 Advantages Easy to calculate and understand Not effected by extreme observations Can be calculated for an open class Median can be found out for qualitative descriptions 24 Disadvantages Does not depend on all observations Some accuracy is given up in choosing a single value to represent the distribution. Not capable of further algebraic treatment Not a good measure for estimation purpose since it is more affected by sampling fluctuations 25 Properties Sum of absolute deviations from median is minimum X |X-med.| |X-7| 4 6 4 2 3 1 8 0 1 10 2 3 12 4 Sum=12 5 Sum=13 26 Quartiles Deciles Percentiles 27 Mode 28 Calculate Mode Classes Below 60 60-62 62-64 64-66 66-68 68-70 70-72 Freq. 12 18 25 30 10 3 3 29 Measures of Variation Also called Dispersion. State the extent to which individual values differ from mean. 30 Significance To determine reliability of an average To serve as a basis for the control of variability To compare two or more series with regard to there variability To facilitate the use of other statistical measures 31 Absolute Variation Relative Variation 32 Methods of studying Variation Range Inter-quartile Range or Quartile deviation Average deviation Standard deviation 33