Define Statictisc and explain it


QUESTION ONE

a)    Define statistics. Using Appropriate Examples, discuss the role of statistics in Educational Administration (12mks)

 

Statistics means a science in which we study how numerical data is to be collected, analyzed and presented and interpreted. The role of statistics in educational administration:-

An aid to supervision

Statistics are an aid to supervision. These are helpful to control the affairs of an organization. The affairs of business organization for example the statistical records maintained are used to evaluate the performance. The management can decide on the basis of this statistics whether the policies are being implemented effectively or not.

Base for planning

Statistics proved a base for the future planning in the absence of relevant data no one can plan properly. Plans are prepared for the expansions of the business in the country by the government. The plans prepared without accurate and relevant data cannot achieve better results.

Eyes Of Administration

Statistics are required by the government to study the causes and find out the remedies of various problems of the country. For example, the government needs adequate statistical data to control crimes to reduce unemployment to reduce the supply conditions of water, electricity etc.

Arithmetic Of Human Welfare

Statistics are used to understand   the problems of human beings. Problems like poverty , food shortage, diseases , illiteracy  etc cannot be understood without  statistical data.

 

Disclose connection between  related factors

Statistics indicate a connection between related facts, for example, there s a relationship between prices, demand and supply, similarly, the increase in sales of a company results in higher profits and vice-versa. There is also relationship between the ages of husband and wives.

Helpful in Business

All types of business decisions are made on the basis of future estimates and expectations. The success of a business   depends upon this fact that to what extent the estimates about future are made accurately statistics is also used in the fields of banking, insurance and transport etc.

Used in all sciences

Statistics are used in all social and natural sciences. Most of the laws of these sciences are proved by the help of statistics. The use of statistics in economics, sociology, physics, chemistry etc is very common.

Helpful In Data Presentation

Statistics is a form of data processing, away of converting data into information useful for decision-making. Processing of raw data is extensively required in the application of many statistical techniques statistical theory is generally expressed in the form of mathematical equations. However the application of this theory requires processing of real data.

b)    Explain the properties  of a good measure of central tendency (8mks)

Measures of central tendency are numbers that define the location of a distribution centre. For example if we regard all measurements as being attempts to give us the ‘true” value of a particular phenomenon, we can regard the value of the distribution of a set of measurements an estimate of that “true” value. Of a particular phenomenon, we ca regard the value of the distribution of a set of measurements an estimate of that “true” value. The various sources of error in the measurements process will produce variability. When dealing with ungrouped data, the researchers can several measures of central tendency. These include the mean, the median and mode when dealing with grouped data, the researcher cannot use the arithmetic mean, instead he/she can use the group mean. Using grouped data the researcher cannot use the median, but can define the modal class.

MEAN: – It is the average. It is found by the sum total divided by the number

MEDIAN: – Is the middle value of the entire distribution

MODE: – It is the value that occurs most often with  certain provisions .

QUESTION TWO (2)

The following data relates to the marks scored by students in an examination.

30        45        58        41        45        78        25        69        32        36

45        69        57        78        65        74        36        25        85        45

78        59        61        62        75        72        68        52        54        50

53        69        51        54        70        63        58        45        68        57

54        52        69        67        86        53        69        67

Required:-

a)     Prepare a group frequency table

b)     Determine:-

  1. Mean
  2. Mode
  3. Standard deviation
  4. Semi-inter-quartile range
  5. The kurtosis
  6. Interpret your results in each case.
Class f Mid Point(X) fx C.F x2 fx2
20-29 2 24.5 49.0 2 600.25 1200.50
30-39 4 34.5 138.0 6 1190.25 4761.00
40-49 6 44.5 267.0 12 1980.25 1188.50
50-59 14 54.5 763.0 26 2970.25 41583.50
60-69 13 64.5 585.0 39 4160.25 54083.25
70-79 7 74.5 521.5 46 5550.25 38851.75
80-89 2 84.5 169.0 48 7140.25 14280.50
  ∑f=48   ∑fx=2746 179 ∑x2=23591.75 ∑fx2=1666.42

 

 

2

Standard Deviation

∑fx2     –     ∑fx

∑f                ∑f

 

2

= 1666.42   –   2746

 

2

48                  48

= 3471.71  –  57.20

= 3471.71-3271.84

= 199.87

 

199.87

S2= 14.14

a)    Prepare a group frequency table

Class Tally Remarks Frequency
20-29 II 2
30-39 IIII 4
40-49 IIIII  I 6
50-59 IIIII  IIIII  IIII 14
60-69 IIIII  IIIII  III 13
70-79 IIIII  II 7
80-89 II 2
    48

 

b)     i) mean=   ∑fx  =2746

∑f        48

= 57.21 the performance of the class is average

ii) Mode = L +      D1                =               D1    = 14-6 =8

D1 + D2            =             14-13=1

 

50 + 8    = 50 + 8

8+1               9

= 50+ 0.89

= 50.89 most of the pupils in the class scored 50.89 Marks

 

iii) S2=                199.87

= 14.14 the variation in performance is less between the top and  the least student.

Iv) Semi-Inter-Quartile Range

=          SIR      = Q3-Q1

2

The value of Q1 = 24.5 + (45-2) x 10

6

= 24.5 + 43 x 10

6

= 24.5 + 430                       = 24.5 + 71.67

6

  = 96.17

 

The upper quartile = N+1    =    179+1       =          180     =          45

4                    4                               4

 

The value of Q3= 54.5 + (45.0+26) x 10

13

= 54.5 + 710

13

=54.5 + 54.61

=109.11

 

The semi-inter-quartile range

=   Q3  –  Q1

2

= 109.11 – 96.17

2

= 12.94

v) Percentile measure of kurtosis

K (Kappa) ½(Q3  –  Q1)

Pqo-P10

 

½(109.11-96.17)

Pqo-P10

 

½       (12.94)

Pqo-P10

Where Q1       –           1st Quartile

Q3       –           3rd Quartile

P10     –           10th Quartile

P90     –           90th Quartile

 

P1= Ln + nN-CF   = nN=  48  = 12

F                       4

= 12

 

QUESTION THREE (3)

a) Using relevant examples , example the difference between primary and secondary data (5mks)

                    i.            Primary data (source

–          Primary data is information gathered directly from respondents. For example the tool like questionnaires, interviews, focused group discussions, observation and experienced studies are used.

–          I it involves creating “new” data. Data is collected from existing sources in an experimental study the variable of interest is identified

 

                 ii.            Secondary Data (Source).

–          This is secondary information sources are data neither collected directly by the user nor superficially for the user. It involves gathering data that has already been collected by someone else. This involves the collection and analysis of published material and information from internal sources. Secondary data collection can be conducted by collecting information from a diverse of documents or electronically stored information.

 

b) Write short notes on the following clearly indicate the relevance of each in statistical analysis.

i)                  Histogram  (3mks)

Histogram is a graph that represents the class frequency in a frequency distribution by vertical rectangles. This consists of a series of rectangles having abase measure along the x-axis proportional to class interval ad an area proportional to frequency. Where the class intervals area equals, the height of the rectangles are proportional to the frequencies. Where the class intervals are not equal, the frequencies are reduced according to ratio between different class intervals and the results are known as frequencies density. Histograms are used to find the value of the mode graphically.

Draw a histogram from the following data

Wages (Sh)                           No. Of Workers

0-10                                                    15

10-20                                                  17

20-30                                                  19

30-40                                                  25

40-50                                                  16

50-60                                                  15

60-70                                                  13

70-80                                                  10

80-90                                                  5

90-100                                                3

 

 

 

ii)                Ogive  Curve(3mks)

An ogive curve is used to find out the values of Median, Quartiles, Deciles and Percentiles graphically .

Example

Class                                          Frequency

0-10                                                    5

10-20                                                  10

20-30                                                  15

30-40                                                  8

40-50                                                  7

 

Solution

Class                                          Frequency             C.F

0-10                                                    5                      5

10-20                                                  10                    15

20-30                                                  15                    30

30-40                                                  8                      38

40-50                                                  7                      45

 

iii)               Bar chart (3marks)

–          Data is represented by a series of bars

Bar charts may be of the following kinds

a)     Simple bar charts-data are represented by a series of bars. The height or length of each bar indicates the size of the figure represented. Numbers of bars depend on the number of figures.

b)     Component bar charts. They are also referred to as subdivided bar charts  into component parts

c)      Multiple bar charts. In this types of charts, the component figure are shown as separate bar charts adjacent each other

 

 

iv)              Pie charts (3marks)

This is a circle divided by radial lines into sections so that the area of each section is proportional to the size of the figure represented. Pie chart is particularly useful when it is desired to show the relatives proportions of the figures that area obtained to make up   a single overall total.

Its advantage is, it is useful where it is desired to show the relative proportion of the figures that go to make up a single overall total.

Example

From the following information construct a pie chart

Product                                  Sales (Sh 000’s)

A                                                          200

B                                                          150

C                                                          100

D                                                         150

Total 600

Product                                  Sales (Sh 000’s)

A                                                          200 x 100       = 120o

600

 

B                                                          150 x 100       = 90o

600

 

C                                                          100 x 100       = 60o

600

 

D                                                         150 x 100       = 90o

600

Total 360o

 

 

v)                Stem And Leaf Plots( 3 Marks)

It is similar to histograms, since it shows how many values in a set fall under a certain interval. However, it has even more information. It shows the actual values within the interval.

Example

–           Here is a stem and leave display of the set 10, 14, 19, 22, 25, 28, 31, 33, 39, 39, 40, 40, 40, 41, 44, 45.

Stem                                  Leaves

1                                        0  4  9

2                                        2  5  5  8

3                                        1  3  9  9

4                                        0  0  0  1  4  5

If we draw a line around the stem or leaf displays the result is histogram.

The stems and leaves need not to be single digits, although all leaves should be the same size in order to make the display accurate.

0 4 9  
2 5 5 8  
1 3 9 9
0 0 0 1 4 5

 

 

 

 

 

 

 

 

 

 

 

QUESTION FOUR (4)

a)    Discuss the functions and relevance of a statistics in the day to day business and management environment (10 Marks).

 

FUNCTIONS

i)       Statistics simplifies the data the complicated data and presents them in such a manner that they once become intelligible. The complicated data may be reduced to totals, averages, percentage etc. and presented graphically or diagrammatically. These devices help to understand quickly the significant characteristics of numerical data.

ii)     Statistics increases man’s experience the proper function of statistics is to enlarge individual experience e.g. one has to conduct an investigation and collect the requisite data. These data will enable a person to get more clear and adequate information about that particular problem.

iii) Statistics  furniture a technique of comparison

–          Certain facts may be meaningless until and unless they are capable of being compared with similar facts at other places or other periods of time e.g. we estimate the per capital income of Kenya not estimate for the value of that fact itself. The statistics afford suitable  techniques for comparison

iv)  Statistics  tests the laws of other science

–          Deductive laws several sciences can be easily verified with the relevant statistical data and application of statistical methods. Like an artist statistics renders useful service in presenting an attract picture of the phenomena under investigation. So close study of pictures enables us to interpret conditions of the phenomena under study.

v)    Statistics helps a lot in Policy Making

–          In order  to form certain policies  statistics provides  us with adequate numerical data relevant  to that phenomena e.g. to form  opinion  about our exports  we have to see how much we produce, how much we consume and how much extra we have. All this can be availed through statistics.

 

 

Relevance of Statistics

i)   Planning and Control

The two principle functions of management business activity. To carry out these functions successfully the management must be supplied with accurate and up-to-date information concerning all aspects of organizations.

 

ii)    Relevant Information

With modern methods of data processing. It is often the case that manager has too much information on which to base a decision. If it is so, the management may wish to reduce the data to a single a vent figures is representative of whole field of data or may wish to reduce the data to a diagrammatic in order to ascertain in the trend.

 

iii) Quality Control

Statistical Methods help the manufacturer to check the quality of output more efficiently.

 

iv)  Forecasting

Time series analysis offers a statistical method for using performance figures to help to produce a forecast for the future to help to produce a forecast for the future. This is particularly important for planning production schedules or advertising expenditure based on sales predictions.

 

v)     Auditing

In fact, this only a particular form of qualify control but it merits a particular mention as that branch  of  statistics  most  accountants every invoice during the audit, the  audition will probably  look only at a small sample of invoices. Using sampling theory, the results from the sample can be interpreted to give information about all invoices.

 

vi)  Production Costs

The technique of correlation and regression enables the statistics to determine a relationship between two variables e.g. costs and methods of production advertising and sales.

b)    Citing relevant examples, explain  the relevance and application of statistics in educational administration and planning (10 marks)

v  Designing of survey and experiments

v  Provide tools for prediction and focusing

v  Applicable to a wide variety of academic discipline including natural and social science.

v  Applicable to government and business

v  Used in summarizing or describing collection of data.

v  Collecting and interpreting data for example when researching a particular topic.

 

It’s Application

Quality  control:- usually there  is a quality  control department  in every industry which is charged with the responsibility of ensuring that the products made do meet the   customers standards e.g. Kenya bureau of standards (KBS) is one of the national  institutions which on behalf of the government inspects the various products  to ensure that they do meet the customers specification.

v  Statistics may be used in making or Ordering Economic Order quantities (EOQ). It is important for Human Manager to realize that it is an economic cost if one orders a large quantity of items which have to be stored for too long before they are sold.

Forecasting  statistics is very important for business managers when predicting the future of  a business for example if a given business situation involves a dependent  and  independent  variables one can develop an equation which can be used to predict the output under  certain given conditions.

Human resources management: Statistics may be used in efficient use of human resources for example we find out where management is weak. By compiling the statistics of those who were signing. It may be found useful to analysis  such data to establish the causes of resignation  thus whether  its by frustration  or by choice.

 

 

 

QUESTION FIVE (5)

The following data relates to the advertising expenditure and sales revenues for a particular company over a period of time.

Sales revenue  Kshs ‘000 2000 2500 3200 5400 2100 3900 5000 4500 3365
Advertising expense Kshs ‘000 500 450 650 740 380 750 450 640 970

 

Required

a)     Identifying the dependent and independent  variables(2marks)

Dependent                           Independent

Advertising                            Sales Revenue

(Y)                                                (X)

b)     Plot the above data on a certain plane. Explain the apparent trend (4 marks)

 

 

The apparent trend is uncertain. Doesn’t take any defined direction.

 

 

 

 

 

 

 

c)     Determine the product moment correlation coefficient. Comment (6marks)

x y xy x2 y2
2000 500

100,000

4,000,000 250,000
2500 450

1,125,000

6,250,000 243,000
3200 650

2,080,000

10240,000 422,500
5400 740

3,996,000

29,160,000 547,600
2100 380

789,000

4,410,000 144,400
3900 750

2,925,000

15,210,000 562,500
5000 450

2,250,000

25,000,000 243,000
4500 640

2,880,000

20,250,000 409,600
3365 970

3,264,050

11,323,225 940,900
∑x=31,965 ∑y=5530

∑xy=19,418,050

∑x2=116,643,225 ∑y2=3,763,500

 

n=n∑xy   –  ∑x ∑y

(n∑x2-(∑x) 2 )(n∑y2-( ∑x)2)

 

r=9 x 19,418,050- 31,965 x 5530

9 x 116,643,225 – (31,965)2 (9 x 3,763,500 – (5530)2

=2,004,000

 

 

(28,027,800)(3,290,600 )   =   96.0355523

 

= 2,004,000

96.0355523

= 20.867

 

d) Determine the 3 moving averages for both sales revenue and advertising expenses respectively (6 marks)

 

3 moving averages for  sales revenue are:-

+ 500, 700, 2200

+ 1700, 1100

– 500 ,-1135

 

3 moving averages for advertising expenses:-

-50, + 200, +90

– 360, +670, -300

+190, -340

 

 

e) Determine the regression equation for the above data (2 marks)

n=9

∑y=an+b∑x

∑xy=a∑+b∑

 

= 5530= 9a + 31,956b………………………….(i)

=19,418,050=31956a+ 116,643225b……(ii)

 

= 9a    =    5530-31956b

9                             9

a  =           5530-31956b……………………….(iii)

9

Put (iii) into (ii)

=19,418,050 x 9 =31956(5530-31956b)9+ 116,643225b

9

=19,418,050 x 9 =31956(5530-31956b) +116,643225b

 

=(19,418,050)=176,766,450-1,021,761,225b+116,643225b

 

=176,766,450=176,766,450-1,021,761,225b+116,643225b

 

=2,004,000=-1,021,761,225b+116,643225b

 

=2,004,000         b=905,118,000

905,118,000          905,118,000

 

b= 22.4

 

But   a: 5530-31965b

9

a= 5530 – 31965(22.4)

7

= 5530 – 716.016

7

= 4813.984

7

a= 687.712

Regression Equation Becomes

y= 687.712 + 22.4

REFERENCE

  1. Kothari C.R. Quantitative(Techniques 3rd Edition), Vikas Publishing Port Hol 2010 Hazanika, P. Bushes Statistics 3rd Edition 2009 S. Chand Co. Ltd.
  2. N.A Salemi(2011) Statistics Simplified Savanis Book Centre Ltd
  3. W.M. Heper (1991), Statistics Preatice  Hall
  4. Srivastava, U.K, Sheroy, Gr Shorna S.C Quantitative Techniques For Management  Decisions 2nd  Edition New Age International

 

 

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Posted on March 11, 2012, in Categorized and tagged , , , , , , , . Bookmark the permalink. Leave a comment.

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