SAMPLING ( INTRODUCTION ) FOR STATISTICS GCE O LEVEL/IGCSE

INTRODUCTION

Sampling is a statistical technique which is used in almost every field in order to collect information and on the basis of this information inferences (results) about the characteristics of a population are made. The numerical values (e.g. mean, standard deviation etc.) calculated from the population is called parameter. And the numerical values which is calculated from the samples is called statistic.

POPULATION

A statistical population (or universe) is defined as the aggregate or totality of all individual members or objects, of some characteristics of interest. For example, the population of a city, the number of students in a school, the number of items in a lot, etc. the individual members of the population are called sampling unit or simply units.

SAMPLE

Sample is the small part chosen from the population, having the same characteristics as population. Sample is a small part which is representing the population. For example, if we want to check the quality of rice in a sack of 100 Kg, we take a small part from it and check the quality of rice.

SAMPLING

It is the procedure in which we select the sample from a population. The two basic purposes of sampling are (i) to provide sufficient information about the characteristics of a population, and (ii) to find the reliability of the estimates derived from the sample.

ADVANTAGES OF SAMPLING

1. Sampling save money as it is much cheaper to collect the information from sample then from population.
2. Sampling saves a lot of time and energy.
3. Sampling provides information that is almost as accurate as that obtained from a complete census.
4. The results of the causes inquiring are sometimes checked on sample basis.
5. In certain circumstances, because of characteristics of the universe, it is only method that can be used.
6. Sampling gives us detailed information about the population.
7. Sampling is extensively used to obtain some of the census information.
8. The most important advantage of sampling is that it provides a valid measure of reliability for the sample estimates.

DISADVANTAGES OF SAMPLING

1. If the sample is not representative of the universe, correct inferences cannot be drawn.
2. Some times the sample may not representative because of reason like inadequate size d sample or wrong method of sampling.
3. When the sampling technique is used, services of experts are necessary.

STANDERDISED VARIABLES ( Z SCORE) FOR STATISTICS GCE O LEVELS/IGCSE

A variable is defined to be Standardized or in standard units if it is expressed in term of deviations from its mean and divided by its standard deviation. It is denoted by Z.

Z=(x-µ)/ δ

The Z-values, being independent of the units of measurement, provided a basis for comparison between individual values, even though they belong to different distributions. That is why the often used in psychological and educational testing, where they are known as standard scores.

Standard Score The standard score is a measure that locates the position of a particular observation with reference to the mean and the standard deviation.

Let x and y be the marks on the original scale and the new scale respectively.

(x-µx)/ δx=(y- µy)/ δy

Where

x is original/unscaled/actual /real value,

y is new/standardised/scaled value,

µx is the actual mean value

µy is the new/Standardized/scaled mean value,

δx is the actual standard deviation,

δy is the new/Standardized/scaled standard deviation.

PROPERTIES OF Z,-SCORE

1. Z-scores are free of units.
2. The mean of Z-score is always zero.
3. The standard deviation of Z-score is always 1.
4. The distribution of Z.-score looks exactly the same as the distribution of the original data.

MEASURE OF CENTRAL TENDENCY ( STANDARD DEVIATION AND VARIANCE) FOR STATISTICS O LEVEL/IGCSE

The variance of a set of observations is defined as the mean of the square of deviations of all the observations from their mean.

Ungrouped formula

Variance = (∑x/n)-(∑x/n)

Frequency Distribution ( grouped )

Variance = (∑ƒx/∑ƒ) – (∑ƒx/∑ƒ)

And the standard deviation or S.D is the square root of the variance

Example: find the variance and S.D of the following set of numbers

1, 3,1,3,4

x	x
1	1
3	9
1	1
3	9
4	16
∑x =12	∑x = 36

Variance= (∑x/n)-(∑x/n)

Variance=(36/5)-(12/5)

Variance=1.44

SD= under root of 1.44

SD=1.2

Example for frequency distribution

Class limits (in miles)	No of employees	X	fx	fx (fx square)
1 – 3	10	2	20	400
4 – 6	14	5	70	4900
7 – 9	10	8	80	6400
10 – 12	6	11	66	4356
13 – 15	5	14	70	4900
16 – 18	5	17	85	7225
	∑F=50		∑fx=391	∑fx=28181

Variance = (∑ƒx/∑ƒ) – (∑ƒx/∑ƒ)

Variance=(28181/50)-(391/50)

Variance= 502.47

SD=under root of 502.47

SD=22.42

MEASURE OF CENTRAL TENDENCY ( STANDARD DEVIATION AND VARIANCE) FOR STATISTICS O LEVEL/IGCSE

The variance of a set of observations is defined as the mean of the square of deviations of all the observations from their mean.

Ungrouped formula

Variance = (∑x/n)-(∑x/n)

Frequency Distribution ( grouped )

Variance = (∑ƒx/∑ƒ) – (∑ƒx/∑ƒ)

And the standard deviation or S.D is the square root of the variance

SD=under root of variance

Example: find the variance and S.D of the following set of numbers

1, 3,1,3,4

x	x
1	1
3	9
1	1
3	9
4	16
∑x =12	∑x = 36

Variance= (∑x/n)-(∑x/n)

Variance=(36/5)-(12/5)

Variance=1.44

SD= under root of 1.44

SD=1.2

Example for frequency distribution

Class limits (in miles)	No of employees	X	fx	fx
1 – 3	10	2	20	400
4 – 6	14	5	70	4900
7 – 9	10	8	80	6400
10 – 12	6	11	66	4356
13 – 15	5	14	70	4900
16 – 18	5	17	85	7225
	∑F=50		∑fx=391	∑fx=28181

Variance = (∑ƒx/∑ƒ) – (∑ƒx/∑ƒ)

Variance=(28181/50)-(391/50)

Variance= 502.47

SD=under root of 502.47

SD=22.42