STATISTICS

Definitions ,[object Object],[object Object],[object Object],[object Object]

SCOPE OF STATISTICS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

SCOPE OF STATISTICS contd… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

SCOPE OF STATISTICS contd… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

SCOPE OF STATISTICS contd… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

SCOPE OF STATISTICS contd… ,[object Object],[object Object],[object Object],[object Object],[object Object]

LIMITATIONS OF STATISTICS ,[object Object],[object Object],[object Object],[object Object]

ROLE OF STATISTICS IN MANAGEMENT DECISIONS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

ROLE OF STATISTICS IN MANAGEMENT DECISIONS contd… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

ROLE OF STATISTICS IN MANAGEMENT DECISIONS contd… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

ROLE OF STATISTICS IN MANAGEMENT DECISIONS contd… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

ROLE OF STATISTICS IN MANAGEMENT DECISIONS contd… ,[object Object],[object Object],[object Object],[object Object],[object Object]

Definitions Continued ,[object Object]

Some More Definitions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

More Definitions ,[object Object]

Illustration – Individual Series ,[object Object],[object Object]

Illustration – Discrete Frequency Distribution 4 68 6 66 10 64 18 62 12 60 No. of Students Height (in inches)

Illustration – Grouped or Continuous Frequency Distribution ,[object Object],9 40-45 23 35-40 40 30-35 2 25-30 8 20-25 Frequency Class-Intervals

Illustration – Grouped or Continuous Frequency Distribution contd… ,[object Object],12 41-50 15 31-40 10 21-30 6 11-20 2 1-10 Frequency Class-Intervals

CONVERSION OF INCLUSIVE TYPE CLASS-INTERVALS TO EXCLUSIVE TYPE CLASS INTERVALS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

CONVERSION OF INCLUSIVE TYPE CLASS-INTERVALS TO EXCLUSIVE TYPE CLASS INTERVALS contd… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],12 41-50 15 31-40 10 21-30 6 11-20 2 1-10 Frequency Class-Intervals

CONVERSION OF INCLUSIVE TYPE CLASS-INTERVALS TO EXCLUSIVE TYPE CLASS INTERVALS contd… ,[object Object],[object Object],12 40.5-50.5 15 30.5-40.5 10 20.5-30.5 6 10.5-20.5 2 0.5-10.5 Frequency Class-Intervals

Obtaining Cumulative Frequency Distribution 73 + 2 = 75 2 2 45-50 65 + 8 = 73 2 + 8 = 10 8 40-45 55 + 10 = 65 10 + 10 = 20 10 35-40 49 + 6 =55 20 + 6 = 26 6 30-35 15 +34 =49 26 + 34 = 60 34 25-30 15 60 + 15 = 75 15 20-25 Less than type More than type Cum.frequency cum.frequency Frequency Class -Intervals

Introduction to Measures of Central Tendency ,[object Object],[object Object],[object Object],[object Object]

Objectives of Averages ,[object Object],[object Object],[object Object],[object Object],[object Object]

Requisites of a Good Average ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Common Measures of Central Tendency ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Averages Mathematical Averages Positional Averages A.M G.M H.M Median Mode

Arithmetic Mean ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Arithmetic Mean contd… ,[object Object],[object Object],[object Object],[object Object],[object Object],f 4 f 4 x 4 x 4 f 3 f 3 x 3 x 3 f 2 f 2 x 2 x 2 f 1 f 1 x 1 x 1 Freq. fx X

μ = 3144 / 50 = 62.88 Ans. Illustration 4 272 50 = N 3144 = Σ fx 68 6 396 66 10 640 64 18 1116 62 12 60 x 12 = 720 60 No. of Students f fX Height (in inches) X

Arithmetic Mean contd… ,[object Object],[object Object],[object Object],[object Object]

Arithmetic Mean Formulae ,[object Object],μ = f 1 x 1 + f 2 x 2 + …..f n x n = Σ fx N Σ f Where N = f 1 +f 2 +…+f n x = mid value of a C.I = (U.L + L.L) 2

Arithmetic Mean Formulae contd… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Arithmetic Mean Formulae contd… ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Illustration – Direct Method ,[object Object],[object Object],μ = Σ fx Σ f 442 = Σ fx 50 = Σ f Total 65 13 5 12-14 110 11 10 10-12 153 9 17 8-10 84 7 12 6-8 30 5 6 4-6 fX Mid-Value X Freq f C.I

Illustration – Assumed Mean Method ,[object Object],[object Object],[object Object],[object Object],μ = A + Σ fd Σ f Σ fd = 105 Σ f= 36 60 15 37.5 4 35-40 60 10 32.5 6 30-35 40 5 27.5 8 25-30 0 0 22.5 = A 9 20-25 -35 -5 17.5 7 15-20 -20 -10 12.5 2 10-15 fd d =(x-A) Mid Values (x) Freq. f C.I

Illustration- Step Deviation Method ,[object Object],[object Object],[object Object],[object Object],μ = A + Σ fd x i Σ f Σ fd = 2100 Σ f= 3600 1200 3 37.5 400 35-40 1200 2 32.5 600 30-35 800 1 27.5 800 25-30 0 0 22.5 = A 900 20-25 -700 -1 17.5 700 15-20 -400 -2 12.5 200 10-15 fd d= (x-A) I (i= 5) MidValues (x) Freq.(f) C.I

Illustration Proceed as usual 2-0=2 90-100 2 90 0 100 9-2=7 80-90 9 80 25-9=16 70-80 25 70 45-25=20 60-70 45 60 75-45=30 50-60 75 50 100-75=25 40-50 100 40 118-100=18 30-40 118 30 133-118=15 20-30 133 20 140-133= 7 10-20 140 10 Freq. C.I Cum. Freq. Marks X or more

What if… ? N=40 Total 3 0-9 15 10-19 10 20-29 8 30-39 3 40-49 1 50-59 Frequency C.I

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Determining missing frequency when A.M is known – Illustration Mean = 16.82 Σ fd = -12 N = 70 + f 4 24 3 32.5 8 30-35 20 2 27.5 10 25-30 14 1 22.5 14 20-25 0 0 17.5 = A ? = f 4 15-20 -16 -1 12.5 16 10-15 -24 -2 7.5 12 5-10 -30 -3 2.5 10 0-5 fd d= (x –A)/i M.V (x) Freq. Marks

Determining missing frequency when A.M is known - Illustration ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Some More Applications of A.M ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Combined A.M ,[object Object],[object Object],[object Object]

Illustration- Combined A.M ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Practice Questions- Arithmetic Mean ,[object Object],[object Object],[object Object],400 600 800 900 700 200 No.of workers 45-50 40-45 35-40 30-35 25-30 20-25 Weekly Income (in Rs.) 2 6 14 18 12 5 3 No.of Students 60-64 55-59 50-54 45-49 40-44 35-39 30-34 Weight (in kgs) 4 10 12 35 25 10 8 No.of workers 425-475 375-425 325-375 275-325 225-275 175-225 125-175 Wages(in Rs.)

Practice Questions- Arithmetic Mean contd… ,[object Object],374 Less than 1000 392 Less than 1100 400 Less than 1200 324 Less than 900 265 Less than 800 194 Less than 700 116 Less than 600 60 Less than 500 20 Less than 400 0 Less than 300 No. of tubes Lifetime (in hrs.)

Merits of A.M ,[object Object],[object Object],[object Object],[object Object],[object Object]

Drawbacks of A.M ,[object Object],[object Object],[object Object],[object Object]

Weighted Arithmetic Mean ,[object Object],[object Object],[object Object]

Illustration – Weighted A.M 106000 = Σ wX 350 = Σ w 15000 150 100 Lower Staff 25000 100 250 Clerical Staff 35000 70 500 Subordinate Staff 16000 20 800 Class II officers 15000 10 1500 Class I Officers wX No. of employees (w) Monthly Salary (in Rs.) (X) Designation

Illustration – Weighted A.M ,[object Object],[object Object],[object Object],[object Object],[object Object]

Median – Positional Average ,[object Object],[object Object]

Calculation of Median ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Calculation of Median – Illustration (Individual Series) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Calculation of Median – Illustration (Individual Series) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Calculation of Median ,[object Object],[object Object],[object Object],[object Object],[object Object]

Calculation of Median-Illustration (Discrete Freq. Distribution) Here N = 50 (i) N/2 = 25 (ii) Cum. Frequency just greater than N/2 = 30 (iii)Corresponding value of item is 62. Median = 60 Ans. 12 12 60 30 18 62 40 10 64 46 6 66 50 4 68 N = 50 Cum. Freq. No. of students Height (in inches)

Calculation of Median ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Md = L1 + N/2 - C (L2 – L1) f

Calculation of Median-Illustration (Grouped Freq. Distribution) N/2 = 3600/2 = 1800 Cum.freq. just greater than 1800 is 2600. Hence median class is 25-30. Hence L1 = 25 L2 = 30 C = 1800 f = 800 Md = 25 + 1800 - 1800 (30 – 25 ) 800 = 25 Ans. Σ f= 3600 3600 400 35-40 3200 600 30-35 2600 800 25-30 1800 900 20-25 900 700 15-20 200 200 10-15 Cum. Freq Freq.(f) C.I

Calculation of Missing Frequencies when median is known : Illustration : Median = 50 N = 100 15 56 + f1 + f2 80-100 ? = f 2 41+ f 1 +f 2 60-80 27 41 + f 1 40-60 ? = f 1 14 + f 1 20-40 14 14 0-20 No. of Families Cumulative Freq. Expenditure

Calculation of Missing Frequencies when median is known : Illustration ,[object Object],[object Object],[object Object],[object Object],Md = L 1 + N/2 - C (L 2 – L 1 ) f 50 = 40 + 50 – (14 + f 1 ) (60 – 40) 27 10 = 720 – 20 f 1 27 f 1 = 450/20 = 22.5 = 23 families approx. N = 56 + f 1 + f 2 100 = 56 + 23 + f 2 f 2 = 21 Ans. f 1 = 23 and f 2 = 21

Practice Numericals - Median ,[object Object],[object Object],7 55-60 13 50-55 15 45-50 20 40-45 30 35-40 33 30-35 28 25-30 14 20-25 No. of Persons Age 125 Less than 80 120 Less than 70 112 Less than 60 96 Less than 50 76 Less than 40 40 Less than 30 16 Less than 20 4 Less than 10 Frequency Value

Practice Problems- Median ,[object Object],229 = N 18 70-80 25 60-70 ? 50-60 65 40-50 ? 30-40 30 20-30 12 10-20 Frequency Class-Intervals

Merits - Median ,[object Object],[object Object],[object Object],[object Object]

Demerits - Median ,[object Object],[object Object],[object Object]

Mode ,[object Object],[object Object],[object Object],[object Object]

Mode – Discrete Frequency Distribution ,[object Object],[object Object],10 Total 1 141 1 140 2 135 2 132 3 130 1 120 No.of students Wt. in pounds

Mode – Continuous Frequency Distribution ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Mode: Formula for Continuous Frequency Distribution Mode = L1 + h(f1 – f0) 2f1-f0-f2

Empirical Relationship between Mean, Median & Mode Mode = 3 Median – 2 Mean

Geometric Mean ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Geometric Mean ,[object Object],[object Object]

Harmonic Mean ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Harmonic Mean ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Harmonic Mean ,[object Object],[object Object]

STATISTICS

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STATISTICS