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# The mean values

## 1.

1## 2. Part 1 THE MEAN VALUES

2## 3. СHAPTER QUESTIONS

1. Measures of location2. Types of means

3. Measures of location for ungrouped data

- Arithmetic mean

- Harmonic mean

- Geometric mean

- Median and Mode

4. Measures of location for grouped data

- Arithmetic mean

- Harmonic mean

- Geometric mean

- Median and Mode

3

## 4.

• Properties to describe numerical data:– Central tendency

– Dispersion

– Shape

• Measures calculated for:

– Sample data

• Statistics

– Entire population

• Parameters

4

## 5.

Measures of location include:• Arithmetic

mean

• Harmonic mean

• Geometric mean

• Median

• Mode

5

## 6.

UNGROUPED or raw data refers to data asthey were collected, that is, before they are

summarised or organised in any way or form

GROUPED data refers to data summarised in

a frequency table

6

## 7. What is the mean?

• The mean - is a generalindicator characterizing the

typical level of varying trait

per unit of qualitatively

homogeneous population.

7

## 8.

• Statistics derive the formula of the means ofthe formula of mean exponential:

X Z

X

Z

n

We introduce the following definitions

- X-bar - the symbol of the mean

Х1, Х2...Хn – measurement of a data value

f- frequency of a data values;

n – population size or sample size.

8

## 9.

• There are the following types ofmean:

• If z = -1 - the harmonic mean,

• z = 0 - the geometric mean,

• z = +1 - arithmetic mean,

• z = +2 - mean square,

• z = +3 - mean cubic, etc.

9

## 10.

• The higher the degree of z, the greater thevalue of the mean. If the characteristic

values are equal, the mean is equal to this

constant.

• There is the following relation, called the

rule the majorizing mean:

x harm x geom x arith x sq

10

## 11.

There are two ways ofcalculating mean:

• for ungrouped data is calculated as a simple mean

• for grouped data is calculated weighted mean

11

## 12. Types of means

MeanFormula

for ungrouped data simple

Harmonic

mean

x

(xf = M)

Geometric

mean

Arithmetic

mean

n

1

x

i

x П ( xi )

n

x

xi

n

for grouped data –

weighted

M

x

M

x

i

i

i

fi

fi

x

П ( xi )

xf

x

f

i i

i

## 13. Arithmetic mean

Arithmetic mean value iscalled the mean value of the

sign, in the calculation of the

total volume of which feature

in the aggregate remains

unchanged

13

## 14. Characteristics of the arithmetic mean

The arithmetic mean has a number ofmathematical properties that can be used to

calculate it in a simplified way.

1. If the data values (Xi) to reduce or increase

by a constant number (A), the mean,

respectively, decrease or increase by a

same constant number (A)

( x A) f

f

i

i

i

x f

f

i

i

i

A f i

f

i

x A

14

## 15.

• 2. If the data values (Xi) divided or multipliedby a constant number (A), the mean

decrease or increase, respectively, in the

same amount of time (this feature allows you

to change the frequency of specific gravities relative frequency):

• a) when divided by a constant number:

xi

1

f

A i A xi f i 1 x

x

fi

fi A A

• b) when multiplied by a constant number:

xAf

f

i

i

A xi f i

f

i

A x

15

## 16.

• 3. If the frequency divided by aconstant number, the mean will

not change:

fi

xi A

fi

A

1

xi f i

A

1

fi

A

x f

f

i

i

x

i

16

## 17.

• 4. Multiplying the mean for the amount offrequency equal to the sum of

multiplications variants on the frequency:

• If

xf

x

f

i i

i

• then the following equality holds:

x fi xi fi

17

## 18.

5.The sum of the deviations of thenumber in a data value from the

mean is zero:

(x x) 0

i

• If xi f i x f i

• then xi f i x f i 0

• So xi f i x f i ( xi x ) f i 0

18

## 19. Measures of location for ungrouped data

• In calculating summary values for a datacollection, the best is to find a central, or

typical, value for the data.

• More important measures of central

tendency are presented in this section:

• Mean (simple or weighter)

• Median and Mode

19

## 20.

ARITHMETIC MEAN- This is the most commonly used measure.

- The arithmetic mean is a summary value

calculated by summing the numerical data

values and dividing by the number of values

sum of sample observations

Sample mean =

number of sample observations

n

x

x

i 1

n

i

Sample size

20

## 21.

ARITHMETIC MEAN- This is the most commonly used measure and

is also called the mean.

sum of observations

Population mean =

number of observations

N

Mean

xi

i 1

N

Xi = observations of the population

∑ = “the sum of”

Population size

21

## 22. Example - The sales of the six largest restaurant chains are presented in table

CompanyMcDonald’s

Sales ($ million)

14.110

Burger King

Kentucky Fried Chicken

Hardee’s

5.590

3.700

3.030

Wendy’s

Pizza Hut

2.800

2.450

A mean sales amount of 5.280 $ million is computed

using Equation of arithmetical mean simple

14100 5590 3700 3030 2800 2450

x

5280

22

6

## 23. MEDIAN for ungrouped data

• The median of a data is the middle item ina set of observation that are arranged in

order of magnitude.

• The median is the measure of location

most often reported for annual income

and property value data.

• A few extremely large incomes or

property values can inflate the mean.

23

## 24.

Characteristics of the median• MEDIAN

– Every ordinal-level, interval-level and ratio-level

data set has a median

– The median is not sensitive to extreme values

– The median does not have valuable mathematical

properties for use in further computations

– Half the values in data set is smaller than median.

– Half the values in data set is larger than median.

– Order the data from small to large.

24

## 25. Position of median

– If n is odd:• Median item number = (n+1)/2

– If n is even:

• Calculate (n+1)/2

• The median is the average of the

values before and after (n+1)/2.

25

## 26. Example

• The median number of people treated daily at theemergency room of St. Luke’s Hospital must be

determined from the following data for the last six

days:

25, 26, 45, 52, 65, 78

Since the data values are arranged from lowest to

highest, the median be easily found. If the data

values are arranged in a mess, they must rank.

Median item number = (6+1)/2 =3,5

Since the median is item 3,5 in the array, the third

and fourth elements need to be averaged:

(45+52)/2=48,5. Therefore, 48,5 is the median

number of patients treated in hospital emergency

26

room during the six-day period.

## 27.

MODE for ungrouped data– Is the observation in the data set that occurs the

most frequently.

– Order the data from small to large.

– If no observation repeats there is no mode.

– If one observation occurs more frequently:

• Unimodal

– If two or more observation occur the same number

of times:

• Multimodal

– Used for nominal scaled variables.

– The mode does not have valuable mathematical

properties for use in future computations

27

## 28.

Example – Given the following data sample:2

5

8

−3

5

2

6

5

−4

The simple mean of the sample of nine

measurements is given by:

9

x

x

i 1

i

n

x21 x52 x83 x−34 x55 x26 x67 x58 x−49

9n

26

2,89

9

28

## 29.

Example – Given the following data set:2

5

8

−3

5

2

6

5

−4

The median of the sample of nine measurements

Odd number

is given by:

−4

−3

2

2

5

5

5

6

8

1

2

3

4

5

6

7

8

9

Median item number =

(n+1)/2 = (9+1)/2 = 5th measurement

Median = 5

29

## 30.

Given the following data set:2

5

8

−3 5

2

6

5

−4 3

Determine the median of the sample of ten measurements.

Order the measurements

Even number

−4

−3

2

2

3

5

5

5

6

8

1

2

3

4

5

6

7

8

9

10

(n+1)/2 = (10+1)/2 = 5,5th measurement

Median = (3+5)/2 = 4

30

## 31.

ExampleGiven the following data set:

2

5

8

−3

5

2

6

5

−4

Determine the mode of the sample of nine measurements.

•Order the measurements

−4

−3

2

2

5

5

5

6

8

Mode = 5

•Unimodal

31

## 32.

ExampleGiven the following data set:

2

5

8

−3

5

2

6

5

−4

2

Determine the mode of the sample of ten measurements.

•Order the measurements

−4

−3

2

2

2

5

5

5

6

8

Mode = 2 and 5

•Multimodal - bimodal

32

## 33.

Harmonic mean for ungrouped data• Is used if М = const:

M

x

M

x

nM

1

M

x

n

1

x

• Harmonic mean is also called the simple

mean of the inverse values .

## 34.

Harmonic mean for ungrouped data• For example:

• One student spends on a solution of task

1/3 hours, the second student – ¼

(quarter) and the third student 1/5 hours.

Harmonic mean will be calculated:

1 1 1

3

3 1

x

(hour )

4

1

1

1

1 3 5 4 12

x 1 1 1

3

5

4

n

## 35. Geometric mean for ungrouped data

• This value is used as theaverage of the relations between

the two values, or in the ranks of

the distributions presented in the

form of a geometric progression.

35

## 36.

Geometric mean for ungrouped datax x1 x2 ... xn П ( xi )

n

n

• Where П – the multiplication of the data

value (Xi).

• n – power of root

## 37.

Geometric mean for ungrouped dataFor example, the known data about the rate

of growth of production

Year

2009

Growth rate 1,24

2010

1,39

2011

1,31

2012

1,15

Calculate the geometric mean. It is 127 percent:

X 1.24 *1.39 *1.31*1.15 1,27

4

## 38.

• ARITHMETIC MEAN– Data is given in a frequency table

– Only an approximate value of the mean

fx

x

f

i

i

i

where f i frequency of the i th class interval

xi = class midpoint of the i th class interval

38

## 39. Example

There are data on seniority hundredemployees in the table

Seniority,

year (х)

The number of

employees (f)

xf

1

9

11

13

15

17

Total

2

10

10

50

20

10

100

3

90

110

650

300

170

1320

39

## 40.

• Average seniority employee is:xf

x

f

1320

13,2 year

100

40

## 41. Harmonic mean for grouped data

• Harmonic mean - is thereciprocal of the arithmetic

mean. Harmonic mean is used

when statistical information

does not contain frequencies,

and presented as

xf = M.

41

## 42. Harmonic mean for grouped data

• Harmonic mean is calculated bythe formula:

M

x

M

x

i

i

i

• where M = xf

42

## 43. Example

There are data on hárvesting the apples bythree teams and on average per worker

Number of

Harvesting the apples, kg

teams

One worker

Whole team

(X)

(M)

1

800

2400

2

1200

9600

3

900

5600

Всего

х

17600

17600

x

1023(kg)

2400 9600 5600

800 1200 900

## 44.

Geometric mean for grouped datais calculated by the formula:

fi

fi

x

П ( xi )

fi

f1

f2

f2

( x ) * ( x ) * ... * ( x )

1

2

2

• Where fi – frequency of the data value (Xi)

П – multiplication sign.

## 45.

Geometric mean for grouped dataEXAMPLE

Year

2010

Growth rate 1,24

2011

1,24

2012

1,31

2013

1,31

Calculate the geometric mean. It is 127,5%

percent:

Х 1.24 2 *1.31 2 1,275

4

## 46.

• MEDIAN– Data is given in a frequency table.

– First cumulative frequency ≥ n/2 will indicate the

median class interval.

– Median can also be determined from the ogive.

M e li

ui li n2 Fi 1

where li

ui

Fi -1

fi

fi

= lower boundary of the median interval

= upper boundary of the median interval

= cumulative frequency of interval foregoing

median interval

= frequency of the median interval

46

## 47.

• MODE– Class interval that has the largest

frequency value will contain the

mode.

– Mode is the class midpoint of this

class.

– Mode must be determined from the

histogram.

47

## 48.

• Mode is calculated by the formula:f Mo f Mo 1

Mo xMo i

f Mo f Mo 1 f Mo f Mo 1

• where хМо – lower boundary of the modal interval

• i= хМо – xMo+1 - difference between the lower

boundary of the modal interval and upper boundary

• fMo, fMo-1, fMo+1 – frequencies of the modal interval,

of interval foregoing modal interval and of interval

following modal interval

48

## 49.

Example – The following data represents the number oftelephone calls received for two days at a municipal call centre.

The data was measured per hour.

Number of

Number of

calls

hours fi

xi

To calculate the

3

3,5

mean for the sample [2–under 5)

[5–under 8)

4

6,5

of the 48 hours:

11

9,5

Determine the class [8–under 11)

[11–under 14)

13

12,5

midpoints

[14–under 17)

9

15,5

[17–under 20)

6

18,5

[20–under 23)

2

21,5 49

n = 48

## 50.

Example – The following data represents the number oftelephone calls received for two days at a municipal call centre.

The data was measured per hour.

Number of

Number of

xi

f i xi

x

calls

hours fi

f

i

[2–under 5)

3

3,5

597

[5–under 8)

4

6,5

48

[8–under 11)

11

9,5

12, 44

[11–under 14)

13

12,5

Average number

[14–under 17)

9

15,5

of calls per hour

[17–under 20)

6

18,5

is 12,44.

[20–under 23)

2

21,5 50

n = 48

## 51.

Example – The following data represents the number oftelephone calls received for two days at a municipal call centre.

The data was measured per hour.

To calculate the for

Number of

Number of

the sample median

calls

hours fi

F

of the 48: hours:

[2–under 5)

3

3

determine the

[5–under 8)

4

7

cumulative

[8–under 11)

11

18

frequencies

[11–under 14)

13

31

[14–under 17)

9

40

n/2 = 48/2 = 24

[17–under 20)

6

46

The first cumulative

[20–under 23)

2

48 51

frequency ≥ 24

n = 48

## 52.

Example – The following data represents the number oftelephone calls received for two days at a municipal call centre.

The data was measured per hour.

Number of

Number of

Median

calls

hours fi

F

ui li n2 Fi 1

li

[2–under 5)

3

3

fi

14 11 24 18 [5–under 8)

4

7

11

13

[8–under 11)

11

18

12,38

[11–under 14)

13

31

50% of the time less

[14–under 17)

9

40

than 12,38 or 50% of [17–under 20)

6

46

the time more than

[20–under 23)

2

48 52

12,38 calls per hour.

n = 48

## 53.

Example – The following data represents the number oftelephone calls received for two days at a municipal call centre.

The data was measured per hour.

The median can

be determined

form the ogive.

Number of calls at a call centre

Number of hours

48

40

32

n/2 = 48/2 = 24

24

16

8

0

2

5

8

11

A

14

17

20

23

Median = 12,4

Read at A.

Number of calls

53

## 54.

Example – The following data represents the number oftelephone calls received for two days at a municipal call centre.

The data was measured per hour.

Number of

Number of

calls

hours fi

To calculate the for

the sample mode

[2–under 5)

3

of the 48 hours

[5–under 8)

4

[8–under 11)

11

The modal interval

[11–under 14)

13

[14–under 17)

9

[17–under 20)

6

The highest

[20–under

23)

2

54

frequency

n = 48

## 55.

MODE• We substitute the data into the formula:

f Mo f Mo 1

Mo xMo i

f Mo f Mo 1 f Mo f Mo 1

13 11

11 (14 11)

12,3

13 11 13 9

• Mo = 12,3

• So, the most frequent number of calls per

hour = 12.3

55

## 56.

Example – The following data represents the number oftelephone calls received for two days at a municipal call centre.

The data was measured per hour.

The mode can

be determined

form the

histogram.

Number of calls at a call centre

Number of hours

14

12

10

8

Mode = 12,3

Read at A.

6

4

2

0

2

5

8

11

17

A 14

Number of calls

20

23

56

## 57. Relationship between mean, median, and mode

• If a distribution is symmetrical:– the mean, median and mode are the same

and lie at centre of distribution

• If a distribution is non-symmetrical:

– skewed to the left or to the right

– three measures differ

A positively skewed distribution

(skewed to the right)

Mode

Mean

Median

Mean

Mode

Median

A negatively skewed distribution

(skewed to the left)

Mean

Mode

Median

57

## 58.

58## 59. EXAMPLE

• Consider a study of the hourlywage rates in three different

companies, For

simplicity,

assume that they employ the

same number of employees: 100

people.

59

## 60.

60## 61.

• So we have three 100-elementsamples, which have the same

average value (35) and the

same variability (120). But these

are different samples. The

diversity of these samples can

be seen even better when we

draw their histograms.

61

## 62.

• The histogram for company I (left chart) issymmetric. The histogram for company II

(middle chart) is right skewed. The

histogram for company III (right chart) is left

skewed. It remains for us to find a way of

determining the type of asymmetry

(skewness) and “distinguishing” it from

symmetry.

62

## 63.

POSITIONAL CHARACTERISTICS• Knowing the median, modal and average

values enables us to resolve the problem

regarding the symmetry of the distribution

of the sample. Hence,

- For symmetrical distributions:

x = Me = Mo ,

- For right skewed distributions:

x > Me > Mo

- For left skewed distributions:

63

x < Me < Mo .

## 64.

• We obtain the following relevantindicators (measures) of asymmetry:

• Index of skewness: X Mo ; X Me

• Standardized skewness ratio:

As

X Mo

As

X Me

• Coefficient of asymmetry

1

( X j X )3

As m3 / 3 n

3

64

## 65. Example

Years Workof

ers

service (f)

6-10

10-14

14-18

18-22

15

30

45

10

Total: 100

Calculation

Хi

xf

Σf=F x x x x * f x x

x x f

2

2

## 66.

Years ofservice

Workers

(f)

Calculation

Midpoint

xf

Σ f=F

Хi

6-10

10-14

14-18

18-22

15

30

45

10

8

12

16

20

Всего:

100

14

66

## 67.

Years ofservice

Workers

(f)

Calculation

Midpoint

xf

Σ f=F

Хi

6-10

10-14

14-18

18-22

15

30

45

10

8

12

16

20

120

360

720

200

15

45

90

100

Всего:

100

14

1400

x

67

## 68.

• The weighted arithmetic meanx f

x

f

i i

i

1400

x

14 years

100

68

## 69.

• The mediann / 2 Fi 1

Me li (Ui li )

fi

50 45

Me 14 (18 14)

14.4

45

69

## 70.

• The modef Mo f Mo 1

Mo xMo i

f Mo f Mo 1 f Mo f Mo 1

45 30

Mo 14 (18 14)

15,2

45 30 45 10

70