Chebychev’s Theorem:

Chebychev’s theorem:
                For any number k greater than 1, at least 1 – 1/k2 of the data-values fall within k standard deviations of the mean, i.e., within the interval (`X – kS,`X + kS)
This means that:
a) At least 1-1/22 = 3/4 will fall within 2 standard deviations of the mean, i.e. within the interval
(`X – 2S,`X + 2S).
b) At least 1-1/32=8/9 of the data-values will fall within 3 standard deviations of the mean, i.e. within the interval (`X – 3S,`X + 3S)
Because of the fact that Chebychev’s theorem requires k to be greater than 1, therefore no useful information is provided by this theorem on the fraction of measurements that fall within 1 standard deviation of the mean, i.e. within the interval (X–S,`X+S).
            Next, let us consider the Empirical Rule mentioned above.
FIVE-NUMBER SUMMARY | |
 A five-number summary consists of   X0,Q1, Median, Q3, and XmIt provides us quite a good idea about the shape of the distribution.If the data were perfectly symmetrical, the following would be true:1. The distance from Q1 to the median would be equal to the distance from the median to Q3:THE SYMMETRIC CURVE. The distance from X0 to Q1 would be equal to the distance from Q3 to Xm. 3. The median, the mid-quartile range, and the midrange would all be equal. All these measures would also be equal to the arithmetic mean of the data:On the other hand, for non-symmetrical distributions, the following would be true:1.         In right-skewed distributions the distance from Q3 to Xm greatly exceeds the distance from X0 to Q1. 2. in right-skewed distributions, median < mid-quartile range < midrange:Similarly, in left-skewed distributions, the distance from X0 to Q1 greatly exceeds the distance from Q3 to Xm.Also, in left-skewed distributions, midrange < mid-quartile range < median.Let us try to understand this concept with the help of an example EXAMPLE:  Suppose that a study is being conducted regarding the annual costs incurred by...