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:
v 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.
v 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.
v 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.
v 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.
v 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 viceversa. There is also relationship between the ages of husband and wives.
v 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.
v 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.
v Helpful In Data Presentation
Statistics is a form of data processing, away of converting data into information useful for decisionmaking. 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:
 Mean
 Mode
 Standard deviation
 Semiinterquartile range
 The kurtosis
 Interpret your results in each case.
Class  f  Mid Point(X)  fx  C.F  x^{2}  fx^{2} 
2029  2  24.5  49.0  2  600.25  1200.50 
3039  4  34.5  138.0  6  1190.25  4761.00 
4049  6  44.5  267.0  12  1980.25  1188.50 
5059  14  54.5  763.0  26  2970.25  41583.50 
6069  13  64.5  585.0  39  4160.25  54083.25 
7079  7  74.5  521.5  46  5550.25  38851.75 
8089  2  84.5  169.0  48  7140.25  14280.50 
∑f=48  ∑fx=2746  179  ∑x^{2}=23591.75  ∑fx^{2}=1666.42 

Standard Deviation
∑fx^{2} – ∑fx
∑f ∑f

= 1666.42 – 2746

48 48
= 3471.71 – 57.20
= 3471.713271.84
= 199.87
199.87
S^{2}= 14.14
a) Prepare a group frequency table
Class  Tally Remarks  Frequency 
2029  II  2 
3039  IIII  4 
4049  IIIII I  6 
5059  IIIII IIIII IIII  14 
6069  IIIII IIIII III  13 
7079  IIIII II  7 
8089  II  2 
48 
b) i) mean= ∑fx =2746
∑f 48
= 57.21 the performance of the class is average
ii) Mode = L + D1 = D1 = 146 =8
D1 + D2 = 1413=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) S^{2}= 199.87
= 14.14 the variation in performance is less between the top and the least student.
Iv) SemiInterQuartile Range
= SIR = Q3Q1
2
The value of Q1 = 24.5 + (452) 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 semiinterquartile range
= Q3 – Q1
2
= 109.11 – 96.17
2
= 12.94
v) Percentile measure of kurtosis
K (Kappa) ½(Q3 – Q1)
PqoP10
½(109.1196.17)
PqoP10
½ _{ }(12.94)
PqoP10
Where Q1 – 1^{st} Quartile
Q3 – 3^{rd} Quartile
P10 – 10^{th} Quartile
P90 – 90^{th} Quartile
P1= Ln + nNCF = 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 xaxis 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
010 15
1020 17
2030 19
3040 25
4050 16
5060 15
6070 13
7080 10
8090 5
90100 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
010 5
1020 10
2030 15
3040 8
4050 7
Solution
Class Frequency C.F
010 5 5
1020 10 15
2030 15 30
3040 8 38
4050 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 chartsdata 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 = 120^{o}
600
B 150 x 100 = 90^{o}
600
C 100 x 100 = 60^{o}
600
D 150 x 100 = 90^{o}
600
Total 360^{o}
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 uptodate 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
v 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.
v 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.
v 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)

c) Determine the product moment correlation coefficient. Comment (6marks)
x  y  xy  x^{2}  y^{2} 
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 
∑x^{2}=116,643,225  ∑y^{2}=3,763,500 
n=n∑xy – ∑x ∑y
(n∑x^{2}(∑x)^{ 2 })(n∑y^{2}( ∑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 = 553031956b
9 9
a = 553031956b……………………….(iii)
9
Put (iii) into (ii)
=19,418,050 x 9 =31956(553031956b)^{9}+ 116,643225b
9
=19,418,050 x 9 =31956(553031956b) +116,643225b
=(19,418,050)=176,766,4501,021,761,225b+116,643225b
=176,766,450=176,766,4501,021,761,225b+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: 553031965b
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
 Kothari C.R. Quantitative(Techniques 3^{rd} Edition), Vikas Publishing Port Hol 2010 Hazanika, P. Bushes Statistics 3^{rd} Edition 2009 S. Chand Co. Ltd.
 N.A Salemi(2011) Statistics Simplified Savanis Book Centre Ltd
 W.M. Heper (1991), Statistics Preatice Hall
 Srivastava, U.K, Sheroy, Gr Shorna S.C Quantitative Techniques For Management Decisions 2^{nd} Edition New Age International
Posted on March 11, 2012, in Categorized and tagged Arithmetic mean, Average, Data, Interquartile range, Math, Median, Standard deviation, Statistics. Bookmark the permalink. Leave a comment.
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