MEASURES OF CENTRAL TENDENCY
The central tendency of a variable means a typical value around which other values tend to concentrate; hence this value representing the central tendency of the series is called measures of central tendency or average.
Statistics that measure the average values of data sets. Usual measures include the mean, median, and mode.
Example: Test scores in a classroom cluster around a mean value.
According to Clark, “Average is an attempt to find one single figure to describe whole of figures.”
- Average is defined as an attempt to find one single figure to describe whole figure.
- Average is frequently referred to as a Measure Of Central Tendency.
- Measures of central value are also popularly known as measures of central tendency because its value lies between two extreme values.
- Types of Average-:
1) Arithmetic mean– simple mean &weighted mean
2) Median
3) Mode
4) Geometric mean
5) Harmonic mean
* Arithmetic mean- Arithmetic Mean (X) : The most popular and widely used measure of representing the entire data by one value is known as arithmetic mean. Its value is obtained by adding together all the items and by dividing this total by the number of items.
- Its value is obtained by adding together all the items and by dividing this total by the number of items.
FORMULA OF ARITHMETIC MEAN
- AM= X1+X2+X3…..XN/N or∑x/n
For correcting incorrect value of ARITHMETIC MEAN is – from incorrect ∑x deduct wrong items and add correct items and then divide the correct with nth observation.
The use of median and mode would be better in open end distributions because of the difficulty of ascertaining lower limit & upper l
Properties of Arithmetic mean
1) The sum of the deviations of the items from the arithmetic means is always zero.
2) Mean is characterized as a point of balance i.e.the sum of the positive deviations from it is equal to the sum of the negative deviations from it.
3) The sum of the squared deviations of the items from arithmetic mean is minimum that is less than the sum of the squared deviations of the items from any other value.
4) Combined mean=x1+x2+x3….xn/n1+n2
- Median & Mode is positional average
- Arithmetic mean, harmonic mean, geometric mean and weighted average mean is a mathematical average.
- Uses of mean in index number and in standardized birth & death rate.
READ ALSO:- DATA COLLECTION
MEDIAN-
Median is that value of the variable which divides the group into two equal parts, one part comprising all values greater than, and the other all values less than the median.
The middle value in the distributions.
It is just the 50th percentile value below which 50% of the values in the sample fall.
Median Is Called The Positional Average
If N is odd then median is an actual value with the remainders of the series in two equal parts on either side of it. If N is even the median is a derived figure, half the sum of the two middle values Odd= middle value
Even = n+1/2th
Mathematical property of median is
1) The sum of the deviations of the items from median, ignoring signs is the least.
Uses of median-
in open end distributions, it is more satisfactory measure of the central tendency than the mean.
Most appropriate average dealing with qualitative data.
Quartiles= 4 equal parts,
deciles= 10 equal parts,
percentiles= 100 equal parts.
Median can be determined by graphic method also by OGIVES.
MODE.
A measure of central tendency for a data set, found by identifying the most frequent data item.
A data set can have more than one mode, as well as no mode.
Examples of modes:
{1, 2, 3, 4, 4, 4, 5, 5, 6, 7, 7, 8}
Mode= 4
When there are two or more values having the same maximum frequency one cannot say which is the modal value & hence mode is said to be ill defined. Such a series is also known as bimodal or multi modal.
Where mode is ill defined its value may be ascertained by the formula based upon relationship between mean, median, mode,
Mode= 3median-2mean.
This measure is called the empirical mode.
We can locate mode graphically using histogram and frequency polygon.
Mode is used in open end distributions/ qualitative phenomenon.
Mode is the most meaning measure of central tendency in case of highly skewed or non- normal distribution, as it provides the best indication of the maximum concern.
Relationship among mean, median and mode is,
CALCULATE MEAN,MEDIAN,MODE,RANGE
GEOMETRIC MEAN It is defined as the nth root of the product of n items or values.
Properties of geometric mean are –
1) The product of the values of series will remain unchanged when the value of geometric mean is substituted for each individual value.
2) The sum of the deviations of the logarithms of the original observations above or below the logarithm of the geometric mean is equal.
Uses-
to find average percentage increase in sales, production, population, in construction of index number.
Geometric mean is not computed when there are both negative & positive values in a series or one more of the values is zero.
HARMONIC MEAN
The Harmonic Mean is based on the reciprocals of the numbers averaged, it is defined as the reciprocal of the arithmetic mean of the reciprocal of the individual observation.
HM = N/(1/x1+1/x2+1/x3….1/xn)
Uses –
It is useful for computing the average rate of increase in profits of a concern or average speed at which a journey has been performed or the average price at which an article has been sold.
The rate usually indicates the relation between two different types of measuring units that can be expressed reciprocally.
Weighted harmonic mean= ∑w/∑(w/x)
Relationship among the averages- AM> GM>HM
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