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

MODE (MEASURE OF CENTRAL TENDENCY) STATISTICS FOR GCE O LEVELS/IGCSE

The mode is defined as the value which occurs most frequently in a set of data that is indicating the most common result.

Example: find the mode of the following set of numbers. 146, 167, 164, 157, 171, 167, 182, 197.

Mode = 167

Solution:

MODE FREQUENCY DISTRIBUTION EXAMPLE

Class Intervals	Frequency
10-14	2
15-19	8
20-24	20
25-29	15
30-34	3
35-39	6

Modal Class is 20-24

ADVANTAGES OF MODE

1. It is simply defined and easily calculated. It is very easy to locate.
2. It is not affected by large or small observations.
3. It can be determined for both the qualitative and quantitative data.
4. It is capable of being ascertained graphically

DISADVANTAGES OF MODE

1. It represents the most frequent value and hence it is very often in practice. The arrangement of data is not necessary.
2. It is not based on all observations made.
3. It is often indeterminate and indefinite.
4. It is not capable of lending itself to further statistical treatment. • It cannot be subjected to algebraic treatments.
5. It is used for the study of most popular fashions.
6. It is extremely used by businessmen and commercial managements.

In histogram, frequency curve or polygon, the mode can be located easily. Class interval/boundary with highest frequency is the mode.

EXAMPLE WORKING THROUGH GRAPH

MEASURE OF CENTRAL TENDENCY ( MEDIAN) FOR STATISTICS O LEVEL/IGCSE

The median is defined as a value which divides the data set that have been ordered, into two

equal parts, one part compromising of observations greater than and the other part smaller than it.

Median=(n/2)th value ( for single set of observations)

Median= L+h/f(n/2-C) for frequency distribution

Note: For odd number of observations we use n+1 in the place of n, above mentioned formulas.

Example: find the median

3, 4, 5, 1, 6, 4,1,8

To find the median first we have to arrange it in ascending order

1,1,3,4,4,5,6,8

Solution:

Median=(8/2)th value

Median=4th value

Median=4

Median frequency distribution example

Class Intervals	Class Boundaries	Frequency	CF
16-19	15.5-19.5	2	2
20-23	19.5-23.5	8	10
24-27	23.5-27.5	18	28
28-31	27.5-31.5	15	43
32-35	31.5-35.5	6	49
36-39	35.5-39.5	1	50

Ef=50

Median value= 50/2= 25^th value

Median=L+h/f(n/2-C)

Median=23.5+4/18(25-10)

Median=26.83

Graphical Determination of the Median

1. With the help of cumulativee frequency curve

Median=n/2 where n=sum of all frequencies.

200/2=100th

Median is between 439.5 and 449.5. Median in statistics can also be called Q2 or Median Quartile.

Advantages of Median

1. It is easily calculated and understood
2. It is located even when the values are not capable of quantitative measurement.
3. It is not affected by extreme values
4. It can be located graphically
5. It can be easily located even if the class intervals in the series are unequal

Disadvantages of Median

1. It is not subjected to algebraic treatments.
2. It cannot be used in further statistical treatment
3. It does not have sampling stability
4. It does not take into account the values of all items in the series.

USES

• It is useful in those cases where numerical measurements are not possible.
• It is also useful in those cases where mathematical calculation cannot be made in order to obtain the mean.
• It is generally used in studying phenomenon like skills, honesty, intelligence etc.

MEASURE OF CENTRAL TENDENCY ( MEAN) FOR STATISTICS O LEVEL/IGCSE

INTRODUCTION

The tendency of the observations to cluster in the central part of the data set is called central tendency and the summary value as a measure of central tendency.

TYPES OF AVERAGES

MEAN

It is defined as the value obtained by dividing the sum of all the observations by their number. Formula given below

Mean=sum of all the observations / number of the observations

Or

Mean = ∑xi/n (for single set of observations)

Mean = ∑ƒx / ∑ƒ (for frequency-distributions)

Example: find the mean of the following set of numbers.(single set of observation)

146,164,157,171,167,182.

Solution

Mean=∑xi/n

Mean = (146+164+157+171+167+182)/6

Mean=6

Example:( for frequency distribution)

x	f (frequency)	f(x) multiply
0	5	0
1	10	10
2	5	10
3	10	30
4	5	20
10	2	20
Sum:	37	90

Step 1. Multiply ƒ(x)

Step 2. Sum ƒ, Sum ƒ(x)

Step 3:∑ƒx/∑ƒ : 90/37=2.43

ADVANTAGES OF MEAN

1. It is based on all the observation in the data.
2. It is easy to calculate and comprehend.
3. It is determined for almost every kind of data.
4. It is the best measure to compare two or more series of data.

DISADVANTAGES OF MEAN

1. It is greatly affected by extreme values in the data.
2. It cannot determine for the quantitative data.
3. It cannot be calculated if all the values are not known.

USES

• A common man uses mean for calculating average results.
• It is extensively used in practical statistics.
• Estimates are always obtained by mean.
• Business man uses it to find the cost or profit per unit of article.