Measures of variation

Measures of variation definition that is the difference between highest value and the lowest value. That is a particular range.  In another way we also simplify that Measures of variation means the quantities that show the measure of variation in terms of random variable. For better understand the measure of variation we should have knowledge of random variables.

Random variable is determined by the outcome of the experiment which we have done. We may assign probabilities to the possible values of the random variable. We can also differ random variable in two ways. First one is discrete random variable in which a variable that can take one value from a discrete set of values. And the second one is continuous random variable, which means a variable that can take one value from a continuous range of values.

Measures of variation in statistics, statistics is a branch of mathematics which gives us the tools to deal with large quantities of data and derive meaningful conclusions about the data. To do this, statistics uses some numbers or a measure which describes the general feature contained in the data. In other word using statistics we can summaries large quantities of data by a few descriptive measures.  In this section we describe how any value is carried out from a given set of data. Measures of the variations sometimes are also known as measure of dispersion or measure of spread.

In statistics there are three measures of the variation…

1. The range
2. The standard deviation
3. The variance

The standard deviation is a measures of the variation or measure of dispersion amongst data. Instead of taking absolute deviation from the arithmetic mean we may square each deviation and obtain the arithmetic mean of squared deviations. This gives us the variance of the values. The positive square root of the variance is called the standard deviation of the given values. The standard deviation may be for raw data and grouped data.

The variance means the square of standard deviation. Alternatively square root of variance is equal to standard deviation. The larger the standard deviation, larger will be the variance. The standard deviation is an absolute measure of variation and hence cannot used for comparing variability of two data sets with different means. Therefore such comparisons are done by using a relative measure of variation known as coefficient of variance.

At last we must take measure of variation examples. Suppose, we have a set in which five different numbers so to find measure of variation we have to calculate only range means difference between highest and lowest value in given set of value.

One Sided Limits: The Right and Left hand limits

One Sided Limits

Limit of a function is given by lim (x->a) f(x) = L, that is, as x approaches ‘a’ from the right hand side of a, f(x) = L and also, as x approaches a from the left hand side of a, f(x)=L.

One Sided Limits Examples
Let us consider a function f(x) = -x^2+6 for  0<=x<2 and f(x)=x-1 for  2<= x <= 4. When we graph this function, we can see that the function is discontinuous at x=2, and the two pieces of the function approach different values, we get, lim (x->2)f(x) is undefined. We can say that the limit of f(x) as x approaches 2 from the left is 2, and the limit of f(x) as x approaches from the right is 1. We can write it as, lim (x->2-) f(x) = 2 and lim (x->2+) f(x) = 1. The plus sign indicates ‘from the right side’ and the minus sign indicates ‘from the left side’.

There are two one sided limits, Right and left hand limits. Right hand limit which is defined as lim (x->a+) f(x) = L , if f(x) is as close to L as we want for all x sufficiently closer to a and x is greater than a, without actually letting x equal to a. Left Hand Limit is defined as lim(x->a-)=L, if f(x) is as close to L as we want for all x sufficiently closer to a and x is less than a, without actually letting x equal to a.

Finding One Sided Limits Algebraically
Let us consider a piecewise function f(x)= x+2 for x less than or equal to 6 and f(x) = x^2-1 for x greater than 6. The steps involved in finding one sided limits of the given piecewise function are as follows:

Left Handed limit of f(x) is given by lim (x->6-)f(x), all the values of x on the number line lie to the left of 6 which means  x is less than 6 and hence we consider the function f(x) = x+2. We plug in the value x=6 in the function which gives, 6+2 = 8

Right Handed Limit of f(x) is given by lim(x->6+) f(x),  all the x values considered on the number line lie to the right of 6, so, x is greater than 6 and hence we consider the function f(x) = x^2-1. We shall plug in the value x=6 in the function which gives, (6)^2-1 = 36-1=35
Lim(x->6)f(x) does not exist as the right handed limit is not equal to the left handed limit