02 quanttech-mean-variance

Quantitative Methods

Varsha Varde

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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

Quantitative Methods

Preliminary Analysis of Data

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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

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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

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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: 491Total Number of Sales Execs : 50

• Arithmetic Mean: 491 / 50 = 9.82 Orders

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This Group

This Group of Participants:

Mode of age is years

Median is years,

Arithmetic Mean is years

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Arithmetic Mean - Example

Product Return on Investment (%)

A 10

B 30

C 5

D 20

Total 65

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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.

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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.

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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

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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

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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.

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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.

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Mean Deviation - ExampleOrders: 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

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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

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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

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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

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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

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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

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Downtime Midpoints No. of Days

05 20

15 40

25 20

35 10

45 10

Total 100

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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

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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.

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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

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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

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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

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Variance

• Definition: Variance is the average of the Squares of the Deviations of the Observed Values from the mean.

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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.

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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

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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

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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..

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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

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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

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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.

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Downtime in Minutes No. of Days

00 – 10 20

10 – 20 40

20 – 30 20

30 – 40 10

40 – 50 10

Total 100

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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

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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

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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%)

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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

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• 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.


• 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.


• 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.

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

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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. ..

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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

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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.

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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

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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

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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

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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

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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

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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

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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

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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=8Remarks:• (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

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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

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Measures of Variation

• Sample range

• Sample variance

• Sample standard deviation

• Sample interquartile range

Sample Range

R = largest obs. - smallest obs.

or, equivalently

R = xmax - xmin

Coefficient of Range

CR = largest obs. - smallest obs. -------------- ----------------------------

largest obs. +smallest obs.

or, equivalently

CR = xmax – xmin/ xmax + xmin

Sample Variance

s

x x

n

ii

n

2

2

1

1

Sample Standard Deviation

s s

x x

n

ii

n

2

2

1

1

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• 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?

Sample Interquartile Range

IQR = third quartile - first quartile

or, equivalently

IQR = Q3 - Q1

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

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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

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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

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Measures of Variability

• 2. Mean Absolute Deviation

• MAD = AM of absolute Deviations• Sum of |xi −¯ x| /n =∑I xi- ¯ x I /nExample 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=6Remarks:• (i) MAD is a good measure of variability• (ii) It is difficult for mathematical manipulations

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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

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• 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,

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• 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

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• 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.

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• 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

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• 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

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• 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

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Sample Mean and VarianceFor Grouped Data

• 5 Example: (weight loss data)• Weight Loss Data• class boundaries mid-pt. freq. xf x2f 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

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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

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• Lmo=13

• f1=5

• f2=6

• f=7

• w=4

• Mode=13+{(7-5)/(14-5-6)}X4=13+8/3

=15.67

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Formulas for Quartiles

• [ (N+1)/4-(F+1)]• Q1=Lq + ------------- x W fqWhere, 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 intervalFirst quartile class is that which includes observation no,

(N+1)/4

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Formulas for Quartiles• [ (N+1)/4-(F+1)]• Q1=Lq + ------------- x W fqWhere, 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=4First quartile class is that which includes observation no,

(N+1)/4=6.5Q1=9+[{(6.5 -4)/5 }x 4]=9+2=11

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• [ 3(N+1)/4-(F+1)]• Q3=Lq + --------------------xW fqWhere, 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 intervalThird quartile class is that which includes observation

no.3(N+1)/4

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• [ 3(N+1)/4-(F+1)]• Q3=Lq + --------------------xW fqWhere, 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=4Third quartile class is that which includes observation

no.3(N+1)/4=19.5Q3=17 +[ {(19.5-16)/6}x4]=17+2.33=19.33

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• [ 2(N+1)/4-(F+1)]• Q2=Lq + ------------------ xW fqWhere, 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 intervalSecond quartile class is that which includes observation no.

(N+1)/2

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• [ 2(N+1)/4-(F+1)]• Q2=Lq + ------------------ xW fqWhere, 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=4Second quartile class is that which includes observation no.

(N+1)/2=13Q2=13 +[{(13-9)/7}x4]=13+5.14=18.14

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Empirical mode

• Where mode is ill defined its value may be ascertained by using the following formula

• Mode =3 median-2mean

Technology

02 quanttech-mean-variance