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### 02 quanttech-mean-variance

1. 1. Quantitative Methods Varsha Varde
2. 2. varsha Varde 2 Course Coverage • Essential Basics for Business Executives • Data Classification & Presentation Tools • Preliminary Analysis & Interpretation of Data • Correlation Model • Regression Model • Time Series Model • Forecasting • Uncertainty and Probability • Sampling Techniques • Estimation and Testing of Hypothesis
3. 3. Quantitative Methods Preliminary Analysis of Data
4. 4. varsha Varde 4 Preliminary Analysis of Data Central Tendency of the Data at Hand: • Need to Size Up the Data At A Glance • Find A Single Number to Summarize the Huge Mass of Data Meaningfully: Average • Tools: Mode Median Arithmetic Mean Weighted Average
5. 5. varsha Varde 5 Mode, Median, and Mean • Mode: Most Frequently Occurring Score • Median: That Value of the Variable Above Which Exactly Half of the Observations Lie • Arithmetic Mean: Ratio of Sum of the Values of A Variable to the Total Number of Values • Mode by Mere Observation, Median needs Counting, Mean requires Computation
6. 6. varsha Varde 6 Example: Number of Sales Orders Booked by 50 Sales Execs April 2006 0, 0, 0, 0, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 12, 14, 15, 16, 17, 19, 21, 24, 28, 30, 34, 43 • Mode: 9 (Occurs 5 Times) Orders • Median: 8 (24 Obs. Above & 24 Below) Total Number of Sales Orders: 491 Total Number of Sales Execs : 50 • Arithmetic Mean: 491 / 50 = 9.82 Orders
7. 7. varsha Varde 7 This Group This Group of Participants: Mode of age is years Median is years, Arithmetic Mean is years
8. 8. varsha Varde 8 Arithmetic Mean - Example Product Return on Investment (%) A 10 B 30 C 5 D 20 Total 65
9. 9. varsha Varde 9 Arithmetic Mean - Example • Arithmetic Mean: 65 / 4 = 16.25 % • Query: But, Are All Products of Equal Importance to the Company? • For Instance, What Are the Sales Volumes of Each Product? Are They Identical? • If Not, Arithmetic Mean Can Mislead.
10. 10. varsha Varde 10 Weighted Average - Example Product RoI Sales (Mn Rs) Weight RoI x W A 10 400 0.20 2.00 B 30 200 0.10 3.00 C 5 900 0.45 2.25 D 20 500 0.25 5.00 Total 65 2000 1.00 12.25 Wt. Av.
11. 11. varsha Varde 11 A Comparison • Mode: Easiest, At A Glance, Crude • Median: Disregards Magnitude of Obs., Only Counts Number of Observations • Arithmetic Mean: Outliers Vitiate It. • Weighted Av. Useful for Averaging Ratios • Symmetrical Distn: Mode=Median=Mean • +ly Skewed Distribution: Mode < Mean • -ly Skewed Distribution: Mode > Mean
12. 12. varsha Varde 12 Preliminary Analysis of Data Measure of Dispersion in the Data: • ‘Average’ is Insufficient to Summarize Huge Data Spread over a Wide Range • Need to Obtain another Number to Know How Widely the Numbers are Spread • Tools: Range & Mean Deviation Variance & Standard Deviation Coefficient of Variation
13. 13. varsha Varde 13 Range and Mean Deviation • Range: Difference Between the Smallest and the Largest Observation • Mean Deviation: Arithmetic Mean of the Deviations of the Observations from an Average, Usually the Mean.
14. 14. varsha Varde 14 Computing Mean Deviation • Select a Measure of Average, say, Mean. • Compute the Difference Between Each Value of the Variable and the Mean. • Multiply the Difference by the Concerned Frequency. • Sum Up the Products. • Divide by the Sum of All Frequencies. • Mean Deviation is the Weighted Average.
15. 15. varsha Varde 15 Mean Deviation - Example Orders: SEs |Orders-Mean| |Orders-Mean| x SEs 00: 04 9.82 9.82x4=39.28 01: 01 8.82 8.82x1=8.82 02: 03 7.82 7.82x3=23.46 03: 03 6.82 6.82x3=20.46 04: 03 5.82 5.82x3=17.46 05: 03 4.82 4.82x3=14.46 06: 04 3.82 3.82x4=15.28 07: 03 2.82 2.82x3=8.46 08: 04 1.82 1.82x4=7.28 09:05 0.82 0.82x5=4.10 10: 02 0.18 0.18x2=0.36
16. 16. varsha Varde 16 Mean Deviation - Example Orders: SEs |Orders-Mean| |Orders-Mean| x SEs 11: 03 1.18 1.18x3=3.54 12: 01 2.18 2.18x1=2.18 13: 00 3.18 3.18x0=0 14: 01 4.18 4.18x1=4.18 15: 01 5.18 5.18x1=5.18 16: 01 6.18 6.18x1=6.18 17: 01 7.18 7.18x1=7.18 18: 00 8.18 8.18x0=0 19: 01 9.18 9.18x1=9.18 20:00 10.18 10.18x0=0 21: 01 11.18 11.18x1=11.18
17. 17. varsha Varde 17 Mean Deviation - Example Orders: SEs |Orders-Mean| |Orders-Mean| x SEs 22: 00 12.18 12.18x0=0 23: 00 13.18 13.18x0=0 24: 01 14.18 14.18x1=14.18 25: 00 15.18 15.18x0=0 26: 00 16.18 16.18x0=0 27: 00 17.18 17.18x0=0 28: 01 18.18 18.18x1=18.18 29: 00 19.18 19.18x0=0 30: 01 20.18 20.18x1=20.18 31: 00 21.18 21.18x0=0 32: 00 22.18 22.18x0=0
18. 18. varsha Varde 18 Mean Deviation - Example Orders: SEs |Orders-Mean| |Orders-Mean| x SEs 33: 00 23.18 23.18x0=0 34: 01 24.18 24.18x1=24.18 35: 00 25.18 25.18x0=0 36: 00 26.18 26.18x0=0 37: 00 27.18 27.18x0=0 38: 00 28.18 28.18x0=0 39: 00 29.18 29.18x0=0 40: 00 30.18 30.18x0=0 41: 00 31.18 31.18x0=0 42: 00 32.18 32.18x0=0 43: 01 33.18 33.18x1=33.18
19. 19. varsha Varde 19 Mean Deviation • Sum of the Products: 318.12 • Sum of All Frequencies: 50 • Mean Deviation: 318.12 / 50 = 6.36 • Let Us Compute With a Simpler Example
20. 20. varsha Varde 20 Machine Downtime Data in Minutes per Day for 100 Working Days Frequency Distribution Downtime in Minutes No. of Days 00 – 10 20 10 – 20 40 20 – 30 20 30 – 40 10 40 – 50 10 Total 100
21. 21. varsha Varde 21 Machine Downtime Data in Minutes per Day for 100 Working Days Frequency Distribution Downtime Midpoints No. of Days 05 20 15 40 25 20 35 10 45 10 Total 100
22. 22. varsha Varde 22 Arithmetic Mean Downtime Midpoints No. of Days Product 05 20 05 x 20 = 100 15 40 15 x 40 = 600 25 20 25 x 20 = 500 35 10 35 x 10 = 350 45 10 45 x 10 = 450 Total 100 2000
23. 23. varsha Varde 23 Arithmetic Mean • Arithmetic Mean is the Average of the Observed Downtimes. • Arithmetic Mean= Total Observed Downtime/ total number of days • Arithmetic Mean= 2000 / 100 = 20 Minutes • Average Machine Downtime is 20 Minutes.
24. 24. varsha Varde 24 Mean Deviation Downtime Midpoints No. of Days Deviation from Mean 05 20 |05 – 20| =15 15 40 |15 – 20| = 05 25 20 |25 – 20| = 05 35 10 |35 – 20| = 15 45 10 |45 – 20| = 25 Total 100
25. 25. varsha Varde 25 Mean Deviation Downtime Midpoints No. of Days Deviation from Mean Products 05 20 |05 – 20| =15 15 x 20 = 300 15 40 |15 – 20| = 05 05 x 40 = 200 25 20 |25 – 20| = 05 05 x 20 = 100 35 10 |35 – 20| = 15 15 x 10 = 150 45 10 |45 – 20| = 25 25 x 10 = 250 Total 100 1000
26. 26. varsha Varde 26 Mean Deviation • Definition: Mean Deviation is mean of Deviations (Disregard negative Sign) of the Observed Values from the Average. • In this Example, Mean Deviation is the Weighted Average(weights as frequencies) of the Deviations of the Observed Downtimes from the Average Downtime. • Mean Deviation = 1000 / 100 = 10 Minutes
27. 27. varsha Varde 27 Variance • Definition: Variance is the average of the Squares of the Deviations of the Observed Values from the mean.
28. 28. varsha Varde 28 Standard Deviation • Definition: Standard Deviation is the Average Amount by which the Values Differ from the Mean, Ignoring the Sign of Difference. • Formula: Positive Square Root of the Variance.
29. 29. varsha Varde 29 Variance Downtime Midpoints No. of Days Difference from Mean Square Products 05 20 05 – 20 = -15 225 225 x 20 = 4500 15 40 15 – 20 = - 05 25 25 x 40 = 1000 25 20 25 – 20 = 05 25 25 x 20 = 500 35 10 35 – 20 = 15 225 225 x 10 = 2250 45 10 45 – 20 = 25 625 625 x 10 = 6250 Total 100 14500
30. 30. varsha Varde 30 Variance & Standard Deviation • Variance = 14500 / 100 = 145 Mts Square • Standard Deviation = Sq. Root of 145 = 12.04 Minutes • Exercise: This Group of 65: Compute the Variance & Standard Deviation of age
31. 31. varsha Varde 31 Simpler Formula for Variance • Logical Definition: Variance is the Average of the Squares of the Deviations of the Observed Values from the mean. • Simpler Formula: Variance is the Mean of the Squares of Values Minus the Square of the Mean of Values..
32. 32. varsha Varde 32 Variance (by Simpler Formula) Downtime Midpoints No. of Days Squares Products 05 20 25 25 x 20 = 500 15 40 225 225 x 40 = 9000 25 20 625 625 x 20 = 12500 35 10 1225 1225 x 10 = 12250 45 10 2025 2025 x 10 = 20250 Total 100 54500
33. 33. varsha Varde 33 Variance (by Simpler Formula) • Mean of the Squares of Values = 54500/100 = 545 • Square of the Mean of Values=20x20=400 • Variance = Mean of Squares of Values Minus Square of Mean of Values = 545 – 400 = 145 • Standard Deviation = Sq.Root 145 = 12.04
34. 34. varsha Varde 34 Significance of Std. Deviation In a Normal Frequency Distribution • 68 % of Values Lie in the Span of Mean Plus / Minus One Standard Deviation. • 95 % of Values Lie in the Span of Mean Plus / Minus Two Standard Deviation. • 99 % of Values Lie in the Span of Mean Plus / Minus Three Standard Deviation. Roughly Valid for Marginally Skewed Distns.
35. 35. varsha Varde 35 Machine Downtime Data in Minutes per Day for 100 Working Days Frequency Distribution Downtime in Minutes No. of Days 00 – 10 20 10 – 20 40 20 – 30 20 30 – 40 10 40 – 50 10 Total 100
36. 36. varsha Varde 36 Interpretation from Mean & Std Dev Machine Downtime Data • Mean = 20 and Standard Deviation = 12 • Span of One Std. Dev. = 20–12 to 20+12 = 8 to 32: 60% Values • Span of Two Std. Dev. = 20–24 to 20+24 = -4 to 44: 95% Values • Span of Three Std. Dev. = 20–36 to 20+36 = -16 to 56: 100% Values
37. 37. varsha Varde 37 Earlier Example Orders: SEs Orders: SEs Orders: SEs Orders: SEs 00: 04 11: 03 22: 00 33: 00 01: 01 12: 01 23: 00 34: 01 02: 03 13: 00 24: 01 35: 00 03: 03 14: 01 25: 00 36: 00 04: 03 15: 01 26: 00 37: 00 05: 03 16: 01 27: 00 38: 00 06: 04 17: 01 28: 01 39: 00 07: 03 18: 00 29: 00 40: 00 08: 04 19: 01 30: 01 41: 00 09: 05 20: 00 31: 00 42: 00 10: 02 21: 01 32: 00 43: 01
38. 38. varsha Varde 38 Interpretation from Mean & Std Dev Sales Orders Data • Mean = 9.82 & Standard Deviation = 6.36 • Round Off To: Mean 10 and Std. Dev 6 • Span of One Std. Dev. = 10–6 to 10+6 = 4 to 16: 31 Values (62%) • Span of Two Std. Dev. = 10–12 to 10+12 = -2 to 22: 45 Values (90%) • Span of Three Std. Dev. = 10–18 to 10+18 = -8 to 28: 47 Values (94%)
39. 39. varsha Varde 39 BIENAYME_CHEBYSHEV RULE • For any distribution percentage of observations lying within +/- k standard deviation of the mean is at least ( 1- 1/k square ) x100 for k>1 • For k=2, at least (1-1/4)100 =75% of observations are contained within 2 standard deviations of the mean
40. 40. varsha Varde 40 Coefficient of Variation • Std. Deviation and Dispersion have Units of Measurement. • To Compare Dispersion in Many Sets of Data (Absenteeism, Production, Profit), We Must Eliminate Unit of Measurement. • Otherwise it’s Apple vs. Orange vs. Mango • Coefficient of Variation is the Ratio of Standard Deviation to Arithmetic Mean. • CoV is Free of Unit of Measurement.
41. 41. Coefficient of Variation • In Our Machine Downtime Example, Coefficient of Variation is 12.04 / 20 = 0.6 or 60% • In Our Sales Orders Example, Coefficient of Variation is 6.36 / 9.82 = 0.65 or 65% • The series for which CV is greater is said to be more variable or less consistent , less uniform, less stable or less homogeneous.
42. 42. Coefficient of Variation • In Our Machine Downtime Example, Coefficient of Variation is 12.04 / 20 = 0.6 • In Our Sales Orders Example, Coefficient of Variation is 6.36 / 9.82 = 0.65 • The series for which CV is greater is said to be more variable or less consistent , less uniform, less stable or less homogeneous.
43. 43. Example • Mean and SD of dividends on equity stocks of TOMCO & Tinplate for the past six years is as follows • Tomco:Mean=15.42%,SD=4.01% • Tinplate:Mean=13.83%, SD=3.19% • CV:Tomco=26.01%,Tinplate=23.01% • Since CV of dividend of Tinplates is less it implies that return on stocks of Tinplate is more stable • For investor seeking stable returns it is better to invest in scrips of Tinplate
44. 44. varsha Varde 44 Exercise • List Ratios Commonly used in Cricket. • Study Individual Scores of Indian Batsmen at the Last One Day Cricket Match. • Are they Nominal, Ordinal or Cardinal Numbers? Discrete or Continuous? • Find Median & Arithmetic Mean. • Compute Range, Mean Deviation, Variance, Standard Deviation & CoV. ..
45. 45. varsha Varde 45 Steps in Constructing a Frequency Distribution (Histogram) 1. Determine the number of classes 2. Determine the class width 3. Locate class boundaries 4. Use Tally Marks for Obtaining Frequencies for each class
46. 46. varsha Varde 46 Rule of thumb • Not too few to lose information content and not too many to lose pattern • The number of classes chosen is usually between 6 and15. • Subject to above the number of classes may be equal to the square root of the number of data points. • The more data one has the larger is the number of classes.
47. 47. varsha Varde 47 Rule of thumb • Every item of data should be included in one and only one class • Adjacent classes should not have interval in between • Classes should not overlap • Class intervals should be of the same width to the extent possible
48. 48. varsha Varde 48 Illustration Frequency and relative frequency distributions (Histograms): Example Weight Loss Data 20.5 19.5 15.6 24.1 9.9 15.4 12.7 5.4 17.0 28.6 16.9 7.8 23.3 11.8 18.4 13.4 14.3 19.2 9.2 16.8 8.8 22.1 20.8 12.6 15.9 • Objective: Provide a useful summary of the available information
49. 49. varsha Varde 49 Illustration • Method: Construct a statistical graph called a “histogram” (or frequency distribution) Weight Loss Data class boundaries - tally class rel . freq, f freq, f/n 1 5.0-9.0 3 3/25 (.12) 2 9.0-13.0 5 5/25 (.20) 3 13.0-17.0 7 7/25 (.28) 4 17.0-21.0 6 6/25 (.24) 5 21.0-25.0 3 3/25 (.12) 6 25.0-29.0 1 1/25 (.04) Totals 25 1.00 Let • k = # of classes • max = largest measurement • min = smallest measurement • n = sample size • w = class width
50. 50. varsha Varde 50 Formulas • k = Square Root of n • w =(max− min)/k • Square Root of 25 = 5. But we used k=6 • w = (28.6−5.4)/6 w = 4.0
51. 51. varsha Varde 51 Numerical methods • Measures of Central Tendency 1. Mean( Arithmetic,Geometric,Harmonic) 2 .Median 3. Mode • Measures of Dispersion (Variability) 1. Range 2. Mean Absolute Deviation (MAD) 3. Variance 4. Standard Deviation
52. 52. varsha Varde 52 Measures of Central Tendency • Given a sample of measurements (x1, x2, · · ·, xn) where n = sample size xi = value of the ith observation in the sample • 1. Arithmetic Mean AM of x =( x1+x2+···+xn) / n = ∑ xi /n • 2. Geometric Mean GM of x =(x1.x2.x3…..xn) ^1/n • 3.Weighted Average = (w1.x1+w2.x2+….wn.xn)/(w1+w2+ …wn) =∑wixi /∑wi
53. 53. varsha Varde 53 Example • : Given a sample of 5 test grades (90, 95, 80, 60, 75) Then n=5; x1=90,x2=95,x3=80,x4=60,x5=75 • AM of x =( 90 + 95 + 80 + 60 + 75)/5 = 400/5=80 • GM of x =( 90 *95* 80 * 60 * 75)^1/5 =(3078000000)^1/5=79 • Weighted verage;w1=1,w2=2,w3=2,w4=3,w5=2 WM of x =( 1*90 + 2*95 + 2*80 +3* 60 +2*75)/10 = 770/10=77
54. 54. varsha Varde 54 Measures of Central Tendency • Sample Median • The median of a sample (data set) is the middle number when the measurements are • arranged in ascending order. • Note: • If n is odd, the median is the middle number If n is even, the median is the average of the middle two numbers. • Example 1: Sample (9, 2, 7, 11, 14), n = 5 • Step 1: arrange in ascending order • 2, 7, 9, 11, 14 • Step 2: med = 9. • Example 2: Sample (9, 2, 7, 11, 6, 14), n = 6 • Step 1: 2, 6, 7, 9, 11, 14 • Step 2: med = (7+9)/2=8 Remarks: • (i) AM of x is sensitive to extreme values • (ii) the median is insensitive to extreme values (because median is a measure of • location or position). • 3. Mode • The mode is the value of x (observation) that occurs with the greatest frequency. • Example: Sample: (9, 2, 7, 11, 14, 7, 2, 7), mode = 7
55. 55. varsha Varde 55 Choosing Appropriate Measure of Location • If data are symmetric, the mean, median, and mode will be approximately the same. • If data are multimodal, report the mean, median and/or mode for each subgroup. • If data are skewed, report the median. • The AM is the most commonly used and is preferred unless precluding circumstances are present
56. 56. varsha Varde 56 Measures of Variation • Sample range • Sample variance • Sample standard deviation • Sample interquartile range
57. 57. Sample Range R = largest obs. - smallest obs. or, equivalently R = xmax - xmin
58. 58. Coefficient of Range CR = largest obs. - smallest obs. -------------- ---------------------------- largest obs. +smallest obs. or, equivalently CR = xmax – xmin/ xmax + xmin
59. 59. Sample Variance ( ) s x x n i i n 2 2 1 1 = − − = ∑
60. 60. Sample Standard Deviation ( ) s s x x n i i n = = − − = ∑2 2 1 1
61. 61. varsha Varde 61 • it is the typical (standard) difference (deviation) of an observation from the mean • think of it as the average distance a data point is from the mean, although this is not strictly true What is a standard deviation?
62. 62. Sample Interquartile Range IQR = third quartile - first quartile or, equivalently IQR = Q3 - Q1
63. 63. Quartile Deviation • Q.D =( third quartile - first quartile)/2 = (Q3 - Q1)/2 • (Median -Q.D) to( Median+Q.D) covers around 50% of the observations as economic or business data are seldom perfectly symmetrical • Coefficient of Quartile deviation =( Q3 - Q1)/ Q3 + Q1
64. 64. varsha Varde 64 Measures of Variation - Some Comments • Range is the simplest, but is very sensitive to outliers • Interquartile range is mainly used with skewed data (or data with outliers) • We will use the standard deviation as a measure of variation often in this course
65. 65. varsha Varde 65 Measures of Variability • Given: a sample of size n • sample: (x1, x2, · · ·, xn) • 1. Range: • Range = largest measurement - smallest measurement • or Range = max - min • Example 1: Sample (90, 85, 65, 75, 70, 95) • Range = max - min = 95-65 = 30
66. 66. varsha Varde 66 Measures of Variability • 2. Mean Absolute Deviation • MAD = AM of absolute Deviations • Sum of |xi −¯ x| /n =∑I xi- ¯ x I /n Example 2: Same sample x x−¯ x |x −¯ x| 90 10 10 85 5 560 65 -15 15 7 -5 5 70 -10 10 95 15 15 Totals 480 0 60 • MAD =60/10=6 Remarks: • (i) MAD is a good measure of variability • (ii) It is difficult for mathematical manipulations
67. 67. varsha Varde 67 Measures of Variability • 3. Standard Deviation • Example: Same sample as before (AM of ;x = 80) ;n=6 x x− ¯x (x − ¯x)2 90 10 100 85 5 25 65 -15 225 75 -5 25 70 -10 100 95 15 225 Totals 480 0 700 • Therefore • Variance of x =700 / 5 =140 ; • • Standard deviation of x = square root of 140 = 11.83
68. 68. varsha Varde 68 • Finite Populations • Let N = population size. • Data: {x1, x2, · · · , xN} • Population mean: μ = (x1+x2+………+xN) /N • Population variance: σ2 = (x1− μ)2+ (x2− μ)2+…….+ (xN− μ)2 ------------------------------------------------------------------- N • Population standard deviation: σ = √σ2,
69. 69. varsha Varde 69 • Population parameters vs sample statistics. • Sample statistics: ¯x, s2 , s. • Population parameters: μ, σ2 , σ. • Approximation: s = range /4 • Coefficient of variation (c.v.) = s / ¯x
70. 70. varsha Varde 70 • 4 Percentiles • Using percentiles is useful if data is badly skewed. • Let x1, x2, . . . , xn be a set of measurements arranged in increasing order. • Definition. Let 0 < p < 100. The pth percentile is a number x such that p% of all measurements fall below the pth percentile and (100 − p)% fall above it.
71. 71. varsha Varde 71 • Example. Data: 2, 5, 8, 10, 11, 14, 17, 20. • (i) Find the 30th percentile. • Solution. • (1)position = .3(n + 1) = .3(9) = 2.7 • (2)30th percentile = 5 + .7(8 − 5) = 5 + 2.1 = 7.1
72. 72. varsha Varde 72 • Special Cases. • 1. Lower Quartile (25th percentile) • Example. • (1) position = .25(n + 1) = .25(9) = 2.25 • (2) Q1 = 5+.25(8 − 5) = 5 + .75 = 5.75 • 2. Median (50th percentile) • Example. • (1) position = .5(n + 1) = .5(9) = 4.5 • (2) median: Q2 = 10+.5(11 − 10) = 10.5
73. 73. varsha Varde 73 • 3. Upper Quartile (75th percentile) • Example. • (1) position = .75(n + 1) = .75(9) = 6.75 • (2) Q3 = 14+.75(17 − 14) = 16.25 • Interquartiles. • IQ = Q3 − Q1 • Exercise. Find the interquartile (IQ) in the above example. • 16.25-5.75=10.5
74. 74. varsha Varde 74 Sample Mean and Variance For Grouped Data • 5 Example: (weight loss data) • Weight Loss Data • class boundaries mid-pt. freq. xf x2 f x f • 1 5.0-9.0- 7 3 21 147 • 2 9.0-13.0- 11 5 55 605 • 3 13.0-17.0- 15 7 105 1,575 • 4 17.0-21.0- 19 6 114 2,166 • 5 21.0-25.0- 23 3 69 1,587 • 6 25.0-29.0 27 1 27 729 • Totals 25 391 6,809 • Let k = number of classes. • Formulas. • AM= (x1f1+x2f2+……..+xkfk)/(f1+f2+……+fk)=391/25=15.64 • Variance= 6809/24-(15.64)^2=283,71-244.61=39 • SD=(39)^1/2=6.24
75. 75. varsha Varde 75 mode for grouped data f – f1 • Mode=Lmo + ---------- x w 2f-f1-f2 • Lmo= Lower limit of Modal Class • f1,f2=Frequencies of classes preceding and succeeding modal class • f=Frequency of modal class • w= Width of class interval
76. 76. varsha Varde 76 • Lmo=13 • f1=5 • f2=6 • f=7 • w=4 • Mode=13+{(7-5)/(14-5-6)}X4=13+8/3 =15.67
77. 77. varsha Varde 77 Formulas for Quartiles • [ (N+1)/4-(F+1)] • Q1=Lq + ------------- x W fq Where, Lq=Lower limit of quartile class N= Total frequency F=Cumulative frequency upto quartile class fq= frequency of quartile class w= Width of the class interval First quartile class is that which includes observation no, (N+1)/4
78. 78. varsha Varde 78 Formulas for Quartiles • [ (N+1)/4-(F+1)] • Q1=Lq + ------------- x W fq Where, Lq=Lower limit of quartile class=9 N= Total frequency=25 F=Cumulative frequency upto quartile class=3 fq= frequency of quartile class=5 w= Width of the class interval=4 First quartile class is that which includes observation no, (N+1)/4=6.5 Q1=9+[{(6.5 -4)/5 }x 4]=9+2=11
79. 79. varsha Varde 79 Formulas for Quartiles • [ 3(N+1)/4-(F+1)] • Q3=Lq + --------------------xW fq Where, Lq=Lower limit of quartile class N= Total frequency F=Cumulative frequency upto quartile class fq= frequency of quartile class w= Width of the class interval Third quartile class is that which includes observation no.3(N+1)/4
80. 80. varsha Varde 80 Formulas for Quartiles • [ 3(N+1)/4-(F+1)] • Q3=Lq + --------------------xW fq Where, Lq=Lower limit of quartile class=17 N= Total frequency=25 F=Cumulative frequency upto quartile class=15 fq= frequency of quartile class=6 w= Width of the class interval=4 Third quartile class is that which includes observation no.3(N+1)/4=19.5 Q3=17 +[ {(19.5-16)/6}x4]=17+2.33=19.33
81. 81. varsha Varde 81 Formulas for Quartiles • [ 2(N+1)/4-(F+1)] • Q2=Lq + ------------------ xW fq Where, Lq=Lower limit of quartile class N= Total frequency F=Cumulative frequency upto quartile class fq= frequency of quartile class w= Width of the class interval Second quartile class is that which includes observation no. (N+1)/2
82. 82. varsha Varde 82 Formulas for Quartiles • [ 2(N+1)/4-(F+1)] • Q2=Lq + ------------------ xW fq Where, Lq=Lower limit of quartile class=13 N= Total frequency=25 F=Cumulative frequency upto quartile class=8 fq= frequency of quartile class=7 w= Width of the class interval=4 Second quartile class is that which includes observation no. (N+1)/2=13 Q2=13 +[{(13-9)/7}x4]=13+5.14=18.14
83. 83. varsha Varde 83 Empirical mode • Where mode is ill defined its value may be ascertained by using the following formula • Mode =3 median-2mean