MEDIAN1. Median is the middle value or the average of two middle values when the values are arranged in an order, either ascending or descending.
2. If there are odd number of observations, median is directly the middle number.
3. If there are even number of observations, median is the average of the two middle numbers.
4. For consecutive integers or for equally spaced numbers (AP),
Median \(=\) \(\frac{(First term + Last term) }{ 2}\).
So,
Median = Mean in this case.
Ex.
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i) The following are the marks of 9 students in a class. Find the median 34, 32, 48, 38, 24, 30, 27, 21, 35
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ii) Find the median of the daily wages of ten workers. Rs. 20, 25, 17, 18, 8, 15, 22, 11, 9, 14
Sol.
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i) Arranging the data in ascending order of magnitude, we have 21, 24, 27, 30, 32, 34, 35, 38,48. Since, there are 9, an odd number of items, therefore median is 32.
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ii) Arranging the wages in ascending order of magnitude, we have 8, 9, 11, 14, 15, 17, 18, 20, 22, 25. Since, there are 10 observations, therefore median is the arithmetic mean of 15 and 17. So the median \(= \frac{(15 + 17)}{ 2 }= 16\).
Mode is the value that occurs most frequently in a set of observations
In case more than one value occurs most frequently, all those values are called the modes.Range. It is defined as the difference between the two extreme observations of the distribution.
\(Range = X_{max} – X_{min}\) where \(X_{max}\) is the greatest observation and \(X_{min}\) is the smallest observation of the variable value.
If Range = 0, all the observations are equal.Standard deviation.
It is defined as the positive square root of the A.M. of the squares of the deviations of the given observations from their A.M. If \(X_1, X_2, ........X_N\) is a set of N observations then its standard deviation is given by Standard Deviation
\(σ =\sqrt{\frac{Σd^2}{N}}\)
or \(σ = \sqrt{\frac{1}{N}∑(X-\overline{X})^2}\)
It is a measure of how much each value varies from the mean of all the values.
Less SD implies more consistency, less variation, less spread, more compactness, AND vice versa.If SD = 0, all the observations are equal.The range is always greater than SD, except when all observations are equal when both are equal to 0.Change in respective statistical parametersAttachment:
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GRE SD.jpg [ 168.14 KiB | Viewed 3087 times ]