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Central Tendency & Dispersion

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The concepts of Mean, Median & Mode are briefly discussed along with examples.

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Central Tendency & Dispersion

  1. 1. MEASURES OF CENTRAL TENDENCY & DISPERSION
  2. 2. BirinderSingh,AssistantProfessor,PCTE Ludhiana 2
  3. 3. INTRODUCTION Measures of central tendency are statistical measures which describe the position of a distribution They are also called statistics of location, and are the complement of statistics of dispersion, which provide information concerning the variance or distribution of observations. In the univariate context, the mean, median and mode are the most commonly used measures of central tendency. BirinderSingh,AssistantProfessor,PCTE Ludhiana 3
  4. 4. CENTRAL TENDENCY OR AVERAGE  An average is a single value which represents the whole set of figures and all other individual items concentrate around it.  It is neither the lowest value in the series nor the highest it lies somewhere between these two extremes.  The average represents all the measurements made on a group, and gives a concise description of the group as a whole.  When two are more groups are measured, the central tendency provides the basis of comparison between them. BirinderSingh,AssistantProfessor,PCTE Ludhiana 4
  5. 5. DEFINITION Simpson and Kafka defined it as “ A measure of central tendency is a typical value around which other figures congregate” Waugh has expressed “An average stand for the whole group of which it forms a part yet represents the whole”. BirinderSingh,AssistantProfessor,PCTE Ludhiana 5
  6. 6. FUNCTIONS OF AVERAGE  Brief Description  Helpful in Comparison  Helpful in formulation of policies  Basis of Statistical Analysis  Representation of the Universe BirinderSingh,AssistantProfessor,PCTE Ludhiana 6
  7. 7. CHARACTERISTICS OF A GOOD AVERAGE  Easy to understand  Simplified  Uniquely defined  Represent the whole group or data  Not affected by extreme values  Capable of further algebraic treatment BirinderSingh,AssistantProfessor,PCTE Ludhiana 7
  8. 8. TYPES OF AVERAGES Types of Averages Mathematical Averages Arithmetic Mean (AM) Geometric Mean (GM) Harmonic Mean (HM) Positional Averages Median (M) Mode (Z) BirinderSingh,AssistantProfessor,PCTE Ludhiana 8
  9. 9. ARITHMETIC MEAN  Most popular & widely used measure  It is defined as the value which is obtained by adding all the items of a series and dividing this total by the number of items.  It is also referred as mean and is denoted as 𝑋 where 𝑋 = 𝐸𝑋 𝑁  It is of two types:  Simple Arithmetic Mean  Weighted Arithmetic Mean BirinderSingh,AssistantProfessor,PCTE Ludhiana 9
  10. 10. METHODS TO SOLVE INDIVIDUAL SERIES  Direct Method 𝑋 = 𝐸𝑋 𝑁 ; 𝑋 = Arithmetic Mean, EX = Sum of observations N = No. of observations  Shortcut Method 𝑋 = A + 𝐸𝑑𝑥 𝑁 ; A = Assumed Mean, dx = X-A BirinderSingh,AssistantProfessor,PCTE Ludhiana 10
  11. 11. PRACTICE PROBLEMS Q1: The pocket allowances of 10 students are as: 15, 20, 30, 22, 25, 18, 40, 50, 55, 65. Calculate the A.M. of their allowances using both methods. BirinderSingh,AssistantProfessor,PCTE Ludhiana 11
  12. 12. METHODS TO SOLVE DISCRETE SERIES  Direct Method 𝑋 = Ʃ𝑓𝑋 𝑁 ; 𝑋 = Arithmetic Mean, f = frequency, N = Ʃf  Shortcut Method 𝑋 = A + Ʃ𝑓𝑑𝑥 𝑁 ; A = Assumed Mean, dx = X-A, N = Ʃf . BirinderSingh,AssistantProfessor,PCTE Ludhiana 12
  13. 13. PRACTICE PROBLEMS Q1: Calculate Arithmetic Mean from the following data using both methods: Wages 10 20 30 40 50 No. of Workers 4 5 3 2 5 BirinderSingh,AssistantProfessor,PCTE Ludhiana 13
  14. 14. METHODS TO SOLVE CONTINUOUS SERIES  Direct Method 𝑋 = Ʃ𝑓𝑋 𝑁 ; 𝑋 = Arithmetic Mean, f = frequency, N = Ʃf  Shortcut Method 𝑋 = A + Ʃ𝑓𝑑𝑥 𝑁 ; A = Assumed Mean, 𝑑𝑥 = X-A, N = Ʃf  . Step Deviation Method 𝑋 = A + Ʃ𝑓𝑑𝑥′ 𝑁 x 𝑖 ; A = Assumed Mean, 𝑑𝑥′ = 𝑋 −𝐴 𝑖 , N = Ʃf, i = common factor BirinderSingh,AssistantProfessor,PCTE Ludhiana 14
  15. 15. PRACTICE PROBLEMS Q1: Compute AM by Step Deviation Method: Ans: 25.85 Q2: Compute AM by Step Deviation Method: (Inclusive Series) Ans: 37.83 Marks 0-10 10-20 20-30 30-40 40-50 No. of students 20 24 40 36 20 Size 20-29 30-39 40-49 50-59 60-69 Frequency 10 8 6 4 2 BirinderSingh,AssistantProfessor,PCTE Ludhiana 15
  16. 16. 16 BirinderSingh,AssistantProfessor,PCTE Ludhiana
  17. 17. PRACTICE PROBLEMS Q3: Compute AM by Step Deviation Method: (Cumulative Frequency Series) Ans: 25.81 Q4: Compute AM by Step Deviation Method: (Cumulative Frequency Series) Ans: 6.33 Marks Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 No. of students 5 17 31 41 49 Size More than 0 More than 2 More than 4 More than 6 More than 8 Frequency 30 28 24 18 10 BirinderSingh,AssistantProfessor,PCTE Ludhiana 17
  18. 18. PRACTICE PROBLEMS Q5: Find the missing frequency if mean is 52: Ans: 7 Q6: Find the missing value if mean of the data is 115.86: Ans: 120 Marks 10-20 20-30 30-40 40-50 50-60 60-70 70-80 No. of students 5 3 4 ? 2 6 13 Wages 110 112 113 117 ? 125 128 130 No. of workers 25 17 13 15 14 8 6 2 BirinderSingh,AssistantProfessor,PCTE Ludhiana 18
  19. 19. PRACTICE PROBLEMS Q7: The sum of the deviations of a certain number of items measured from 2.5 is 50 and from 3.5 is -50. Find number of items and mean. Ans: 100, 3 Q8: The mean height of 25 male workers is 61 cms and the mean height of 35 female workers is 58 cms. Find the combined mean height of 60 workers in the factory. (Combined AM) Ans: 59.25 Q9: The mean wage of 100 workers in a factory, running in two shifts of 60 and 40 workers respectively is Rs. 38. The mean wage of 60 workers working in the morning shift is Rs. 40. Find the mean wage of 40 workers working in the evening shift. (Combined AM) Ans: 35 BirinderSingh,AssistantProfessor,PCTE Ludhiana 19
  20. 20. PRACTICE PROBLEMS – TYPICAL EXAMPLES Q10: The mean monthly salary paid to all employees in a certain company was Rs. 600. The mean monthly salaries paid to male & female employees were Rs. 620 and Rs. 520 respectively. Find the percentage of male to female employees in the company. Q11: A bookseller has 150 books of Physics & Chemistry whose average price is Rs. 40 per book. Average price of Physics book is Rs. 43 and that of Chemistry is Rs. 35. Find the number of books of each subject with the book seller. BirinderSingh,AssistantProfessor,PCTE Ludhiana 20
  21. 21. PRACTICE PROBLEMS – TYPICAL EXAMPLES Q12: The mean of 100 items is 80. By mistake one item is misread as 92 instead of 29. Find the correct mean. (Ans: 79.37) Q13: The mean of 5 observations is 7. Later on it was found that two observations 4 and 8 were wrongly taken instead of 5 and 9. Find the correct mean. (Ans: 7.4) Q14: The average daily price of a share of a company from Monday to Friday was Rs. 130. If the highest & lowest price during the week were Rs. 200 and Rs. 100 respectively, find the average daily price when the highest & lowest price are not included. (Ans: 116.67) BirinderSingh,AssistantProfessor,PCTE Ludhiana 21
  22. 22. DISADVANTAGES OF MEAN: • It is affected by extreme values. • It cannot be calculated for open end classes. • It cannot be located graphically • It gives misleading conclusions. • It has upward bias. BirinderSingh,AssistantProfessor,PCTE Ludhiana 22
  23. 23. GEOMETRIC MEAN (GM)  It is defined as the nth root of the product of all the n values of the variable.  GM = 𝑋1. 𝑋2. 𝑋3. … . . 𝑋𝑛 𝑛 where 𝑋1. 𝑋2. 𝑋3. … . . etc. are the various values of the series n = number of items  If there are two items in a series say 2 & 8, then their GM = 2 𝑥 8 2 = 16 2 = 4  If there are three items in a series say 2 ,4 & 8, then their GM = 2 𝑥 4 𝑥 8 3 = 64 3 = 4  If, the number of items in a series is very large, it would be difficult to calculate the geometric mean.  For calculating GM, log values are used. BirinderSingh,AssistantProfessor,PCTE Ludhiana 23
  24. 24. GEOMETRIC MEAN – USING LOGARITHMIC CALCULATION  GM = (𝑋1. 𝑋2. 𝑋3. … . . 𝑋𝑛)1/ 𝑛 Taking log on both sides Log GM = 1 𝑛 (log𝑋1 + 𝑙𝑜𝑔𝑋2 + 𝑙𝑜𝑔𝑋3 + … . . +𝑙𝑜𝑔𝑋𝑛) Log GM = Ʃ 𝑙𝑜𝑔𝑋 𝑛 Taking Antilog on both sides GM = Antilog ( Ʃ 𝑙𝑜𝑔𝑋 𝑛 )  The values of logs and antilog can be obtained from the log table and antilog tables. BirinderSingh,AssistantProfessor,PCTE Ludhiana 24
  25. 25. CALCULATION OF GM – INDIVIDUAL SERIES  Find the logarithms of the given values.  Find the sum total of logs, i.e. Ʃ log X.  Divide Ʃ log X by the number of items (N)  Calculate Antilog of the value  The result will be the Geometric Mean BirinderSingh,AssistantProfessor,PCTE Ludhiana 25
  26. 26. PRACTICE PROBLEMS – INDIVIDUAL SERIES Q1: Calculate Geometric Mean of the following series: i. 180, 190, 240, 386, 492, 662 ii. 2574, 475, 75, 5, 0.8, 0.08, 0.005, 0.0009 iii. 0.9841, 0.3154, 0.0252, 0.0068, 0.0200, 0.0002, 0.5444, 0.4010 Answers: i. 317.9 ii. 1.841 iii. 0.0511 BirinderSingh,AssistantProfessor,PCTE Ludhiana 26
  27. 27. CALCULATION OF GM – DISCRETE SERIES & CONTINUOUS SERIES  Find the logarithms of the given values i.e. log X  Multiply the frequency with the corresponding log values  Find the sum total of logs, i.e. Ʃ f log X.  Divide Ʃ f log X by the total number of frequencies (N = Ʃf)  Calculate Antilog of the value  The result will be the Geometric Mean BirinderSingh,AssistantProfessor,PCTE Ludhiana 27
  28. 28. PRACTICE PROBLEMS – DISCRETE & CONTINUOUS SERIES Q4: Calculate Geometric Mean of the following: (Ans: 8.82) Q5: Calculate Geometric Mean of the following: (Ans: 19.27) X 6 7 8 9 10 11 12 F 8 12 18 26 16 12 8 Marks 0-10 10-20 20-30 30-40 40-50 Students 3 4 6 3 2 BirinderSingh,AssistantProfessor,PCTE Ludhiana 28
  29. 29. COMBINED GEOMETRIC MEAN  If G1 & G2 are the Geometric Means of two groups having N1 & N2 items, then the combined GM is given by the following formula: G12 = Antilog ( 𝑁1 log 𝐺1 +𝑁2 log 𝐺2 𝑁1 +𝑁2 ) Q6: The GM of a sample of 10 items was found to be 20 and that of a sample of 20 items was found to be 15. Find the combined GM. (Ans: 16.51) BirinderSingh,AssistantProfessor,PCTE Ludhiana 29
  30. 30. WEIGHTED GEOMETRIC MEAN  If G1 & G2 are the Geometric Means of two groups having W1, W2, …… weights, then the weighted GM is given by the following formula: G12 = Antilog ( 𝑊1 log 𝐺1 +𝑊2 log 𝐺2 + ….. 𝑊1 +𝑊2 + …….. ) Q6: Calculate weighted GM: (Ans: 119.3) Items Index No. Weights Food 120 7 Rent 110 5 Clothing 125 3 Fuel 105 2 Others 140 3 BirinderSingh,AssistantProfessor,PCTE Ludhiana 30
  31. 31. AVERAGE RATE OF GROWTH OF POPULATION  GM is also used to compute the average annual percent increase in population, prices when the values of the variables at the beginning of the first and at the end of the nth period are given. The average annual percent increase may be computed by applying the formula: Pn = P0 (1+r)n where r = the average rate of growth n = Number of years P0 = Value at the beginning of the period Pn = Value at the end of the period Formula: r = Antilog ( log 𝑃 𝑛 −log 𝑃0 𝑛 ) – 1 BirinderSingh,AssistantProfessor,PCTE Ludhiana 31
  32. 32. PRACTICE PROBLEMS – AVERAGE RATE OF GROWTH OF POPULATION Q7: The population of a country has increased from 84 million in 1983 to 108 million in 1993. Find the annual rate of growth of population. (Ans: 2.6%) Q8: The population of a town was 10000. At first, it increased at the rate of 3% p.a. for the first three years and then it decreased at the rate of 2% p.a. for the next 2 years. What will be the population of the town after 5 years? (Ans: 10494) Q9: The population of a city was 1,00,000 in 1980 and 1,44,000 in 1990. Estimate the population at the middle of 1980-1990. (Ans: 120000) BirinderSingh,AssistantProfessor,PCTE Ludhiana 32
  33. 33. HARMONIC MEAN (H.M.)  It is a mathematical average.  It is based on the reciprocal of items.  It is defined as the reciprocal of the artihmetic average of the reciprocal of the values of irs various items.  HM = 𝑁 Σ ( 1 𝑋 )  It is useful in finding averages involving speed, time, price and ratios. BirinderSingh,AssistantProfessor,PCTE Ludhiana 33
  34. 34. CALCULATION OF HARMONIC MEAN – INDIVIDUAL SERIES  Find out the reciprocals of the values of the series  Add the values of the reciprocals to get Σ( 1 𝑥 )  Divide the number of items by the sum total of reciprocals.  The final value is the Harmonic Mean. Q1: Calculate HM of the following: 2, 4, 7, 12, 19 Ans: 4.86 BirinderSingh,AssistantProfessor,PCTE Ludhiana 34
  35. 35. CALCULATION OF HARMONIC MEAN – DISCRETE & CONTINUOUS SERIES  Divide each frequency by the respective values of the variable.  Obtain the total Σ( 𝑓 𝑥 )  Divide the number of items by the Σ( 𝑓 𝑥 )  The final value is the Harmonic Mean. BirinderSingh,AssistantProfessor,PCTE Ludhiana 35
  36. 36. PRACTICE PROBLEMS – H.M. Q2: The following table gives the marks obtained by students in a class. Calculate the H.M. Ans: 23.24 Q3: Calculate the H.M. for the following: Ans: 20.48 Marks 18 21 30 45 No. of students 6 12 9 2 Marks 0-10 10-20 20-30 30-40 40-50 No. of students 4 7 28 12 9 BirinderSingh,AssistantProfessor,PCTE Ludhiana 36
  37. 37. WEIGHTED HARMONIC MEAN  Weighted H.M. = Σ𝑊 Σ( 𝑊 𝑋 ) Q4: Find the weighted H.M. of the items 4, 7, 12, 19, 25 with weights 1, 2, 1, 1, 1 respectively. BirinderSingh,AssistantProfessor,PCTE Ludhiana 37
  38. 38. APPLICATIONS OF HARMONIC MEAN Q5: An aeroplane covers the four sides of a square field at speeds of 1000, 2000, 3000 and 4000 km/hr respectively. What is its average speed? Ans: 1920 km/hr Q6: A cyclist covers first three kms at an average speed of 8 km/hr, another 2 kms ar 3 km/hr and the last two kms at 2 km/hr. Find the average speed and verify your answer. Ans: 3.42 km/hr Q7: Typist A can type a letter in 5 minutes, B in 10 minutes & C in 15 minutes. What is the average number of letters typed per hour per typist? Ans: 7.33 BirinderSingh,AssistantProfessor,PCTE Ludhiana 38
  39. 39. RELATIONSHIP BETWEEN A.M., G.M. & H.M.  G.M. = 𝐴. 𝑀. 𝑥 𝐻. 𝑀.  When all the values of the series differ in size, A.M. > G.M. > H.M.  When all the values of the series are equal, A.M. = G.M. = H.M. Q8: If AM of two numbers is 10 and their GM is 8, find the two numbers and HM. Ans: 16, 4, 6.4 Q9: Using the values 2, 4 and 8, verify that A.M. > G.M. > H.M. BirinderSingh,AssistantProfessor,PCTE Ludhiana 39
  40. 40. Median is a central value of the distribution, or the value which divides the distribution in equal parts, each part containing equal number of items. Thus it is the central value of the variable, when the values are arranged in order of magnitude. Connor has defined as “ The median is that value of the variable which divides the group into two equal parts, one part comprising of all values greater, and the other, all values less than median” MEDIAN BirinderSingh,AssistantProfessor,PCTE Ludhiana 40
  41. 41. CALCULATION OF MEDIAN – DISCRETE SERIES  Arrange the data in ascending or descending order.  Calculate the cumulative frequencies.  Apply the formula : M = Size of 𝑵+𝟏 𝟐 𝒕𝒉 𝒊𝒕𝒆𝒎  Now locate 𝑁+1 2 th items in the cumulative frequency column. The value to be selected which is equal to or higher than 𝑁+1 2 value.  Median is the value of the variable corresponding to the selected value. BirinderSingh,AssistantProfessor,PCTE Ludhiana 41
  42. 42. PRACTICE PROBLEMS – DISCRETE SERIES Q1: Calculate the median from the following data: X: 10 12 14 16 18 20 22 f: 2 5 12 20 10 7 3 BirinderSingh,AssistantProfessor,PCTE Ludhiana 42
  43. 43. CALCULATION OF MEDIAN – CONTINUOUS SERIES  Arrange the data in ascending order  Calculate the cumulative frequencies.  Find out the median size by using the formula: N/2  Determine the median class in which median lies  Apply the formula : M = l1 + 𝑵 𝟐 −𝒄.𝒇. 𝒇 x i; where l1 = lower limit of the median class c.f. = cumulative frequency of the class preceeding the median class f = frequency of the median class i = size of the class interval of the median class BirinderSingh,AssistantProfessor,PCTE Ludhiana 43
  44. 44. PRACTICE PROBLEMS – CONTINUOUS SERIES Q1: Calculate the median from the following data: (Ans: 17.5) Q2: Calculate Median: (Ans: 36.25) X: 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45 f: 6 12 17 30 10 10 8 5 2 Value Frequency Less than 10 4 Less than 20 16 Less than 30 40 Less than 40 76 Less than 50 96 Less than 60 112 Less than 70 120 Less than 80 125 BirinderSingh,AssistantProfessor,PCTE Ludhiana 44
  45. 45. PRACTICE PROBLEMS – CONTINUOUS SERIES Q3: Calculate the median from the following data: (Inclusive) (Ans: 25) Q4: Calculate Median: (Unequal Class Interval) (Ans: 32.67) Q5: Calculate Median (Descending Class Intervals) (Ans: 18.125) X: 1--10 11-20 21-30 31-40 41-50 f: 4 12 20 9 5 Size 10-15 15-17.5 17.5-20 20-30 30-35 35-40 40 + f: 10 15 17 25 28 30 40 Marks 30-35 25-30 20-25 15-20 10-15 5-10 0-5 Student s 4 8 12 16 10 6 4 BirinderSingh,AssistantProfessor,PCTE Ludhiana 45
  46. 46. PRACTICE PROBLEMS – CONTINUOUS SERIES Q6: Calculate Median: (Mid Value Series) (Ans: 45.385) Q7: Find the missing frequencies if N = 100 & M = 30. (Ans: 15, 10) Q8: Find the missing frequencies if N = 229 & M = 46. (Ans: 34*, 45) X: 5 15 25 35 45 55 65 75 f: 15 7 11 10 13 8 20 16 Size 0-10 10-20 20-30 30-40 40-50 50-60 f: 10 ? 25 30 ? 10 Size 10-20 20-30 30-40 40-50 50-60 60-70 70-80 f: 12 30 ? 65 ? 25 18 BirinderSingh,AssistantProfessor,PCTE Ludhiana 46
  47. 47. PRACTICE PROBLEMS Q9: The median of the few number of observations is 48.6. The four items having values are 35, 36.5, 49.1, 50 were added to given series, what will be the new median? (Ans: 48.6) BirinderSingh,AssistantProfessor,PCTE Ludhiana 47
  48. 48. MERITS OF MEDIAN  Median can be calculated in all distributions.  Median can be understood even by common people.  Median can be ascertained even with the extreme items.  It can be located graphically  It is the most appropriate average in case of open ended classes  It is the most suitable average in dealing with qualitative facts such as beauty, intelligence, honesty etc. BirinderSingh,AssistantProfessor,PCTE Ludhiana 48
  49. 49. DISADVANTAGES OF MEDIAN  It is not based on all the values.  It is not capable of further mathematical treatment like arithmetic mean  It is affected fluctuation of sampling.  In case of even no. of values it may not the value from the data. BirinderSingh,AssistantProfessor,PCTE Ludhiana 49
  50. 50. QUARTILES, DECILES & PERCENTILES  Just as median divides the series into two equal parts, there are other measures which divide the series  Quartiles: These divide a series into 4 equal parts. For any series, there are three quartiles called Q1 (lower – 25%), Q2 (median – 50%), Q3 (upper – 75%)  Deciles: These divide a series into 10 equal parts. For any series, there are 9 deciles denoted by D1, D2, ….. D9.  Percentiles: These divide a series into 100 equal parts. For any series, there are 99 percentiles denoted by P1, P2, ….. P99. BirinderSingh,AssistantProfessor,PCTE Ludhiana 50
  51. 51. CALCULATION OF QUARTILES, DECILES & PERCENTILES For Individual & Discrete Series For Continuous Series Formula to be used in Continuous Series Q1 = Size of 𝑁+1 4 th item Q1 = Size of 𝑁 4 th item Q1 = l1 + 𝑵 𝟒 −𝒄.𝒇. 𝒇 x i Q3 = Size of 3(𝑁+1) 4 th item Q3 = Size of 3𝑁 4 th item Q3 = l1 + 𝟑𝑵 𝟒 −𝒄.𝒇. 𝒇 x i D1 = Size of 𝑁+1 10 th item D1 = Size of 𝑁 10 th item D1= l1 + 𝑵 𝟏𝟎 −𝒄.𝒇. 𝒇 x i D9 = Size of 9(𝑁+1) 10 th item D9 = Size of 9𝑁 10 th item D9 = l1 + 𝟗𝑵 𝟏𝟎 −𝒄.𝒇. 𝒇 x i P1 = Size of 𝑁+1 100 th item P1 = Size of 𝑁 100 th item P1 = l1 + 𝑵 𝟏𝟎𝟎 −𝒄.𝒇. 𝒇 x i P99 = Size of 99(𝑁+1) 100 th item P99 = Size of 99𝑁 100 th item P99 = l1 + 𝟗𝟗𝑵 𝟏𝟎𝟎 −𝒄.𝒇. 𝒇 x i BirinderSingh,AssistantProfessor,PCTE Ludhiana 51
  52. 52. PRACTICE PROBLEMS Q1: From the following data, calculate Q1, Q3, D5, P25 21, 15, 40, 30, 26, 45, 50, 54, 60, 65, 70 Ans: 26, 60, 45, 26 Q2: From the following data, calculate Q1, Q3, D6, P85 Ans: 13,15,14,16 Q3: Calculate Q1, Q3, D8, P56 Ans: 14.7, 34, 36.3, 26.4 X 10 11 12 13 14 15 16 17 18 f 3 4 5 12 10 7 5 2 1 Wages 0-10 10-20 20-30 30-40 40-50 No. of Workers 22 38 46 35 19 BirinderSingh,AssistantProfessor,PCTE Ludhiana 52
  53. 53. PRACTICE PROBLEMS – TYPICAL Q4: The first & third quartiles of the following data are given to be 25 marks and 50 marks respectively out of the given data below: Find out the missing frequencies when N = 72 Ans: 12, 11, 3 Marks Frequency 0-10 4 10-20 8 20-30 ? 30-40 19 40-50 ? 50-60 10 60-70 5 70-80 ? BirinderSingh,AssistantProfessor,PCTE Ludhiana 53
  54. 54. MODE  Mode is the most frequent value or score in the distribution.  It is defined as that value of the item in a series.  It is denoted by the capital letter Z.  It is the highest point of the frequencies distribution curve.  Croxton and Cowden : defined it as “the mode of a distribution is the value at the point armed with the item tend to most heavily concentrated. It may be regarded as the most typical of a series of value” BirinderSingh,AssistantProfessor,PCTE Ludhiana 54
  55. 55. CALCULATION OF MODE – INDIVIDUAL SERIES  Inspection Method: In this method, simply identify the value that occurs most frequently in a series. Q1: Find the mode from the following data: 8, 10, 5, 8, 12, 7, 8, 9, 11, 7 & 7, 8, 10, 15, 10, 22, 20, 26, 20, 34, 20, 6, 10 &, 10, 15, 20, 25, 30 (Ans: 8, Bi Modal, No Mode)  By changing the Individual Series into Discrete Series: First convert the series into discrete, then identify the value corresponding the highest frequency, that value will be mode. Q2: Find the mode from the following data: 8, 10, 5, 8, 12, 7, 8, 9, 11, 7 (Ans: 8) BirinderSingh,AssistantProfessor,PCTE Ludhiana 55
  56. 56. CALCULATION OF MODE – DISCRETE SERIES  Inspection Method: By inspecting, identify the value whose frequency is maximum, is Mode. Q3: Find the mode from the below data: (Ans: 14000) Q4: Find the mode from the below data: (Ans: 7) Income 11000 12000 13000 14000 15000 16000 No. of persons 2 4 7 10 4 3 Marks 5 6 7 8 9 10 No. of students 8 11 15 2 3 1 BirinderSingh,AssistantProfessor,PCTE Ludhiana 56
  57. 57. CALCULATION OF MODE – DISCRETE SERIES  Grouping Method: In some cases, it is possible that value having the highest frequency may not be the modal value.  It happens where the difference between the maximum frequency and the frequency preceeding or succeeding it is very small and items are heavily concentrated on either side.  In such cases, mode can be determined by grouping method.  Here, modal value is determined by preparing two tables – Grouping Table & Analysis Table BirinderSingh,AssistantProfessor,PCTE Ludhiana 57
  58. 58. CALCULATION OF MODE – DISCRETE SERIES Q5: Find the mode using grouping method: (Ans: 12) Q6: Find the mode using grouping method: (Ans: 40) X 7 8 9 10 11 12 13 14 15 16 17 f 2 3 6 12 20 24 25 7 5 3 1 X 20 25 30 35 40 45 50 55 f 1 3 5 9 14 10 6 4 BirinderSingh,AssistantProfessor,PCTE Ludhiana 58
  59. 59. CALCULATION OF MODE – CONTINUOUS SERIES  Firstly, modal class is determined by using inspection method or grouping method, as similar to discrete series  After determining modal class, mode can be found out using the following formula: Z = 𝑙1 + 𝑓1 − 𝑓0 2𝑓1 − 𝑓0 − 𝑓2 𝑥 𝑖 𝑙1 = lower limit of the modal class 𝑓1 = frequency of the modal class 𝑓0 = frequency of the pre modal class 𝑓2 = frequency of the post modal class 𝑖 = size of the modal class BirinderSingh,AssistantProfessor,PCTE Ludhiana 59
  60. 60. CALCULATION OF MODE – CONTINUOUS SERIES  Few Important points to remember:  If the first class is the modal class, the 𝑓0 is taken as zero.  If the last class is the modal class, the 𝑓2 is taken as zero.  If the modal value lies outside the modal class (Failure of Formula), then the following formula is used to calculate the mode: Z = 𝑙1 + 𝑓2 𝑓0 + 𝑓2 𝑥 𝑖  If mode is ill defined (Bi-Modal Series), then we use the formula Z = 3M – 2𝑋 BirinderSingh,AssistantProfessor,PCTE Ludhiana 60
  61. 61. PRACTICE PROBLEMS - MODE Q7: Calculate Mode: Ans: 18 Q8: Calculate Mode: (Cumulative Frequency Series) Ans: 27.73 Wages 0-5 5-10 10-15 15-20 20-25 25-30 30-35 No. of workers 3 7 15 30 20 10 5 Marks between No. of students 10 and 15 4 10 and 20 12 10 and 25 30 10 and 30 60 10 and 35 80 10 and 40 90 10 and 45 95 10 and 50 97 BirinderSingh,AssistantProfessor,PCTE Ludhiana 61
  62. 62. PRACTICE PROBLEMS - MODE Q9: Calculate Mode: (Unequal Class Intervals) Ans: 33.47X f 0-5 4 5-10 8 10-20 10 20-25 9 25-30 13 30-40 30 40-44 6 44-50 9 50-70 12 BirinderSingh,AssistantProfessor,PCTE Ludhiana 62
  63. 63. PRACTICE PROBLEMS – MODE (TYPICAL) Q10: Calculate mode: (Bi-Modal Series) (Ans: 𝑋 = 49.51, M = 49.69, Z = 50.05) Q11: Calculate mode: (Failure of Formula) (Ans: Z = 48.89) Marks 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 No. of students 4 6 20 32 33 17 8 2 Marks 25- 35 35- 45 45- 55 55- 65 65- 75 75- 85 85- 95 95- 105 105- 115 No. of students 4 44 38 28 6 8 12 2 2 BirinderSingh,AssistantProfessor,PCTE Ludhiana 63
  64. 64. PRACTICE PROBLEMS – MODE (TYPICAL) Q12: Calculate mode: (First Class as Modal Class) (Ans: Z = 160) Q13: Calculate mode: (Last Class as Modal Class) (Ans: Z = 163.58) Marks 100-200 200-300 300-400 400-500 500-600 No. of students 27 9 7 3 2 Marks 155-157 157-159 159-161 161-163 163-165 No. of students 4 8 26 53 89 BirinderSingh,AssistantProfessor,PCTE Ludhiana 64
  65. 65. PRACTICE PROBLEMS – MODE (MISSING FREQUENCY) Q14: The median and mode for the distribution are Rs. 25 and Rs. 24 respectively. Find the missing frequencies. (Ans: 25 & 24) Q15: The median and mode of the following wage distribution of 230 workers are known to be Rs. 33.5 and Rs. 34 respectively. Find the missing values: (Ans: 60, 100 & 40) Expenditure 0-10 10-20 20-30 30-40 40-50 No. of families 14 ? 27 ? 15 Wages 0-10 10-20 20-30 30-40 40-50 50-60 60-70 No. of workers 4 16 ? ? ? 6 4 BirinderSingh,AssistantProfessor,PCTE Ludhiana 65
  66. 66. BirinderSingh,AssistantProfessor,PCTE Ludhiana 66
  67. 67. ADVANTAGES OF MODE : • Mode is readily comprehensible and easily calculated • It is the best representative of data • It is not at all affected by extreme value. • The value of mode can also be determined graphically. • It is usually an actual value of an important part of the series. BirinderSingh,AssistantProfessor,PCTE Ludhiana 67
  68. 68. DISADVANTAGES OF MODE :  It is not based on all observations.  It is not capable of further mathematical manipulation.  Mode is affected to a great extent by sampling fluctuations.  Choice of grouping has great influence on the value of mode. BirinderSingh,AssistantProfessor,PCTE Ludhiana 68
  69. 69. CONCLUSION • A measure of central tendency is a measure that tells us where the middle of a bunch of data lies. • Mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers in a set of data. This is also known as average. BirinderSingh,AssistantProfessor,PCTE Ludhiana 69
  70. 70. • Median is the number present in the middle when the numbers in a set of data are arranged in ascending or descending order. If the number of numbers in a data set is even, then the median is the mean of the two middle numbers. • Mode is the value that occurs most frequently in a set of data. BirinderSingh,AssistantProfessor,PCTE Ludhiana 70

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