Statictics and Measures of Central Tendency

STATICTICS AND

MEASURES OF CENTRAL TENDENCY

STATICTICSThe word statistics refers to quantitative information or to a method of dealing with quantitative information.

The methods by which statistical data are analyzed are called STATISTICAL METHODS.

STATISTICAL METHODS are applicable to a very large number of fields-economics,sociology,antropology,business,agriculture,psychology,medicines,education.

FUNCTIONS OF STATISTICSIt presents facts in a definite formIt simplifies mass of figuresIt facilitates comparisonIt helps in formulating and testing hypothesisIt helps in predictionIt helps in the formulation of suitable policies

CHARTING DATAMost appropriate way to represent the data

is through charts.

Pictorial representation helps in quick understanding of the data.

A chart can take the shape of either a diagram or a graph.

A pie chart is a circular representation of

data when a circle is divided into sectors

with areas equal to the corresponding

component. These sectors are called slices

and represent the percentage breakdown of

the corresponding components.

BAR-CHARTBAR-CHARTA bar chart is a graphical device used in depicting data that have been summarized as frequency, relative frequency, or percentage frequency

The statistical meaning of histogram is that it is graph that represent the class frequencies in frequency distribution by vertical adjacent rectangle

Example: The following is a list of prices (in dollars) of birthday cards found in various drug stores:

1.45 2.20 0.75 1.23 1.25 1.25 3.09 1.99 2.00 0.78 1.32 2.25 3.15 3.85 0.52 0.99 1.38 1.75 1.22 1.75

we Make a frequency distribution table for this data.

We omit the units (dollars) while calculating. The values go from 0.52 to 3.85 , which is roughly 0.50 to 4.00 . We can divide this into 7 intervals of equal length: 0.50 - 0.99 , 1.00 - 1.49 , 1.50 - 1.99 , 2.00 - 2.49 , 2.50 - 2.99 , 3.00 - 3.49 , and 3.50 - 3.99 . Then we can count the number of data points which fall into each interval--for example, 4 points fall into the first interval: 0.75, 0.78, 0.55, and 0.99--and make a frequency distribution table:

Intervals in (dollars) frequency0.50- 0.99 41.00- 1.49 7 1.50 -1.99 3 2.00 -2.49 32.50 -2.99 0 3.00 -3.49 2 3.50 -3.99 1Total 20

Making a Histogram Using a Frequency Distribution Table A histogram is a bar graph which shows frequency distribution.To make a histogram, follow these steps: 1.On the vertical axis, place frequencies. Label this axis "Frequency". 2.On the horizontal axis, place the lower value of each interval. Label this axis with the type of data shown (price of birthday cards, etc.) 3.Draw a bar extending from the lower value of each interval to the lower value of the next interval. The height of each bar should be equal to the frequency of its corresponding interval.

Example: Make a histogram showing the frequency distribution of the price of birthday cards. Histogram

A frequency polygon is a graph of frequency distribution. It is particularly effective in comparing two or more frequency distribution.There are two way in which a frequency polygon may be constructed:1. we may draw histogram of the given data and then join by straight lines the mid-point of the upper horizontal side of each rectangle with the adjacent rectangle.2. anther method of constructing frequency polygon is to take the mid-points of the various class-intervals and then plot the frequency corresponding to each point and to join all these points by straight lines. j

# Draw an ogive and determine the no. of companies getting profits between Rs. 45 crore and Rs. 75 crore.

OGIVE

75

PARETO CHART It is a special type of vertical bar chart in which

the categorized responses are plotted in the

descending rank order of their frequencies and

combined with a cumulative polygon on the same

graph.

The main focus of the Pareto chart is to separate

the “vital few” from the “trivial many.”

Case: A departmental store conducted a customer satisfaction survey of 455 frequent customers . The survey group prepared a questionnaire that was divided in two parts: satisfaction reasons and dissatisfaction reasons. Departmental store has decided to focus on the reasons of dissatisfaction to improve its quality of service. The following observations regarding categories of dissatisfaction were made:

Sl. No.

Dissatisfaction reasons No. of customers

123456

Poor product rangeLack of staffsLate Home deliveryParkingMissing price tags No card payment service

12540653075120

Total 455

Pareto chart of the case of department store

STEM AND LEAF PLOT Stem-and-leaf plot can be constructed by separating the digits of

each number into two groups, one as a stem and the other as a leaf.

This plot is mainly used for examining the shape and spread of data.

After separating the data, the left-most digit is termed as the stem and is the higher valued digit. The right-most digit is termed as the leaf and is the lower valued digit.

State of auto manufacturingAccording to data released by the Automotive News Data Center, General MotorsCorporation is number one in

the world in totalvehicle sales of cars and light trucks. FordMotor Company is number two followed by Toyota Motor Corporation

and Volkswagen,respectively. Between 1999 and 2000, while General Motors maintained its number oneposition, it

sold nearly 200,000 fewer cars worldwide. During this same period, Ford Motorincreased sales by more than 200,000. The

greatest percentage growth from 1999 to 2000was for PSA Peugeot-Citroen, which increased sales by 14.2%. The global

sales figures forthe top 10 auto manufacturers of cars and light trucks for both 1999 and 2000 follow.Suppose you are a

business analyst for one of these companies. Your manager asks you toprepare a brief report showing the state of car and light truck

sales in the world. You areto compare your company's position with other firms.

Company 1999 2000 % change

General motors

8786000 8591327 -2.2

Ford motors 7148000 7350495 2.8

Toyota motors 5359000 5703446 6.4

volkswagen 4860203 5161188 6.2

daimlerchrysler

4864500 4749000 2.4

Psa peugeot-citroen

2519600 2877900 14.2

fiat 2521000 2646500 5.0

Hyundai motor

2600862 2634560 1.3

Nissan motor 2567878 2629044 2.4

Honda motor 2395000 2540000 6.1

Q.:Suppose DaimlerChrysler randomly samples 40 dealerships and discovers that

the following data tell how many car and light trucks were sold at these dealerships

last month. How can you summarize these data in a report?34 58 40 49 49 57 44 57 69 45 64 31 47 30 44 44 51 65 60 6561 62 68 43 66 63 44 34 57 44 67 61 47 67 52 34 58

59 45 33

SOLUTION:

A stem and leaf plot of these data would appear as follows.FREQUENCY Stem & Leaf

6.00 3. 0 1 3 4 4 415.00 4. 0 3 4 4 4 4 4 5 5 7 7 9 910.00 5. 1 2 7 7 7 8 8 913.00 6. 0 I I 2 3 4 5 5 6 7 7 8 9

SCATTER PLOTThe scatter plot is a graphical presentation of the

relationship between two numerical variables.

It generally shows the nature of the relationship

between two variables.

The application of a scatter plot is very common

in regression, multiple regressions, correlation,

etc.

Q. How might you graphically depict the 1999 data against the 2000 data?

Company 1999 2000 % change

General motors

8786000 8591327 -2.2

Ford motors 7148000 7350495 2.8

Toyota motors 5359000 5703446 6.4

volkswagen 4860203 5161188 6.2

daimlerchrysler

4864500 4749000 2.4

Psa peugeot-citroen

2519600 2877900 14.2

fiat 2521000 2646500 5.0

Hyundai motor

2600862 2634560 1.3

Nissan motor 2567878 2629044 2.4

Honda motor 2395000 2540000 6.1

Representation through scatter plot

0

2000000

4000000

6000000

8000000

10000000

0 5000000 10000000

SALES VALUES FOR YEAR 1999 AND 2000

X AXIS-1999, Y AXIS-2000

ARITHMETIC MEAN The arithmetic mean (AM) of a set of observations is their

sum, divided by the number of observations. It is generally denoted by x or AM. Population mean is

denoted by μ.

Arithmetic mean is of two types:

Simple arithmetic mean Weighted arithmetic mean

Weighted Arithmetic Mean

The weighted mean enables us to calculate an average that takes into account the importance of each value to the overall total.

GEOMETRIC MEAN Geometric mean (GM) is the nth root of the product of n items of

a series.

Commonly used in the calculation of average rate of growth.

Chemical,industrial & pharmaceutical laboratories(cipla)

Cipla was registered as a public ltd co. with authorized capital of rs.60000 million in 1935.operations officially started in sept 1937.its products & services are categorized as prescription,animal health care products etc.It now exports to countries in europe,america,africa,asia,mid-east,asia.it has won “EXPRESS PHARMA PULSE AWARD” for “sustained growth” for 205-06.It overtook ranbaxy and glaxosmithkline to become the largest pharmaceutical co. in the domestic market in 2007.

Mean= 8631.138889

HARMONIC MEANBased on the reciprocal of the numbers averaged.Defined as the reciprocal of the arithmetic mean of the

reciprocal of the individual observation.

It can be written as

Applications of Harmonic MeanUseful for computing average rates

e.g. Average rate of increase of profits or average speed at which any journey has been performed.

Chemical,industrial & pharmaceutical laboratories(cipla)

Sales turnover for year 1989-2006YEAR SALES

(In million Rs.) (x)

1/X

1989 971.2 0.00102971990 928.9 0.00107651991 1236.4 0.00080881992 1514 0.00066051993 1990.3 0.00050241994 2454.7 0.00040741995 2987.1 0.00033481996 3623.6 0.0002761997 4525.8 0.0002211998 5170.8 0.00019341999 6255.4 0.00015992000 7721.4 0.0001292001 10643.1 0.0000932002 14008.1 0.0000712003 15730.2 0.0000632004 20554.2 0.0000482005 24008.9 0.0000412006 31036.2 0.000032

∑1/x = 0.0061473

n=18∑ 1/x=0.0061473

Harmonic Mean = n

∑ 1/x

= 18/0.0061473

= 2928.1148

Merit and limitationsIt is useful in special cases of averaging

rates.It can not be used when there are both

positive and negative observations or one or more observations have zero values.

It is rarely used in business problems

The median may be defined as the middle or central value of the variable when values are arranged in the order of magnitude.

In other words, median is defined as that value of the variable that divides the group into two equal parts, one part comprising all values greater and the other all values lesser than the median.

To measure the qualitative characteristics of data, other

measures of central tendency, namely median and mode are

used.

Positional averages, as the name indicates, mainly focus on

the position of the value of an observation in the data set.

UNGROUPED DATA:

1) If n(no. of observation) is odd, middle term of a series is size of ((n+1)/2)th term & is the value of a median.

2) If n(no. of observation) is even, middle term of a series are (n/2)th and (n/2+1)th terms . So, arithmetic mean of both the observations is our

median.

GROUPED DATA:

MEDIAN=L+ [N/2-p.c.f]/F* I

Where L=lower limit of class

N=sum of all the frequencies

p.c.f=preceding cumulative frequency to median class

F=frequency of median class

i=class interval of median class

Chemical,industrial & pharmaceutical laboratories(cipla)

Cipla was registered as a public ltd co. with authorized capital of rs.60000 million in 1935.operations officially started in sept 1937.its products & services are categorized as prescription,animal health care products etc.It now exports to countries in europe,america,africa,asia,mid-east,asia.it has won “EXPRESS PHARMA PULSE AWARD” for “sustained growth” for 205-06.It overtook ranbaxy and glaxosmithkline to become the largest pharmaceutical co. in the domestic market in 2007.

SALES TURNOVER FROM 1989-2006

MEDIAN : total observations(n)=18 n/2th term=9th term=4525.8 (n/2+1)th term=10th term=5170.8

median=(4525.8+5170.8)/2 =4848.3

MODE

UNGROUPED DATA:Mode is that value which occurs the maximum no. of times

GROUPED DATA:Mode=L+(f-f1)/(2f-f1-f2) *I

Where, L=lower limit of modal classf = frequency of modal class f 1=frequency of class preceding the modal classf 2=frequency of class succeeding the modal classi=size of modal class

Chemical,industrial & pharmaceutical laboratories(cipla)

Cipla was registered as a public ltd co. with authorized capital of rs.60000 million in 1935.operations officially started in sept 1937.its products & services are categorized as prescription,animal health care products etc.It now exports to countries in europe,america,africa,asia,mid-east,asia.it has won “EXPRESS PHARMA PULSE AWARD” for “sustained growth” for 205-06.It overtook ranbaxy and glaxosmithkline to become the largest pharmaceutical co. in the domestic market in 2007.

SALES TURNOVER FROM 1989-2006 mode=0

RELATIONSHIP BETWEEN MEAN,MEDIAN & MODE

*distribution in which values of mean,median,mode coincide are symmetrical distribution.

*distribution in which values of mean,median,mode are not equal are asymmetrical or skewed.

The distance between mean & median is approximately one-third of the distance between the mean and mode.

Acc. To Karl Pearson :

Mean-median=1/3(mean-mode)

Mode=3median-2mean

Median=(2mean+mode)/3

QUARTILES Partition values are measures that divide the data into several

equal parts. Quartiles divide data into 4 equal parts, deciles divide data into 10 equal parts, and percentiles divide data into 100 equal parts.

For an individual series, the first and third quartiles can be computed using the following formula:

Values of quartiles can be measured as : Q(for k=1 to n)=k(n+1)/4

Example : Q1=first quartile = (n+1)/4

DECILES In a data series, when the observations are arranged

in an ordered sequence, deciles divide the data into 10 equal parts. In the case of individual series and discrete frequency distribution, the generalized formula for computing deciles is given as:

Values of decile can be measured as :D(for k=1 to n)=k(n+1)/10Example: D1=(N+1)/10

PERCENTILES In a data series, when observations are arranged in an ordered

sequence, percentiles divide the data into 100 equal parts. For an individual series and a discrete frequency distribution, the generalized formula for computing percentiles is given as:

VALUES OF PERCENTILE CAN BE MEASURED AS :P(for k=1 to n)=k(n+1)/100Example : P1=(n+1)/100

Example: From the following data, find the first and third quartiles.

The first and third quartiles can be computed by applying the formula

discussed above. The data is already arranged in an ordered manner:

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