The alternative formula for obtaining the sum of squares is: 9
Sum of squares = 2 x - 2 n x The two new expressions are 2 x and 2 x (i) 2 x is called the sum of the squares of x . Using again the 10 measurements of Example 3 it is derived as follows: x : 81 79 82 83 80 78 80 87 82 82 2 x = 81 2 + 79 2 + 82 2 + 83 2 + 80 2 + 78 2 + 80 2 + 872 + 82 2 + 82 2 = 66316 (ii) 2 x is called the square of the sum of x and is calculated as follows: 2 x = (81 + 79 + 82 + 83 + 80 + 78 + 80 + 87 + 82 + 82) 2 = 662596 Substituting in the formula for the sum of squares: Sum of squares = 4 . 56 10 662596 66316 (iii) The variance is obtained by dividing the sum of squares by ( 1 n ): 27 . 6 ) 1 10 ( 4 . 56 2 s (iv) The standard deviation is the square root of the variance: mm s 504 . 2 27 . 6 1.7Degrees of freedom In obtaining a sample standard deviation to estimate a population standard deviation, reference was made to the number of degrees of 10
freedom. Because the concept of degrees of freedom is involved in many statistical techniques, it now needs a fuller explanation. Suppose that we are told that a sample of n = 5 observations has a mean x of 50 and we are then asked to ‘invent’ the values of the observations. We know that x = ( x x n ) = 250. If the sum of the observations is 250, we have freedom of choice only for the first four observations because the 5 th must be a number (perhaps a negative number) that brings the sum to 250. By way of example, if the first four numbers are arbitrarily chosen as, say, 40, 25, 18, 130, then in order to make x = 250, we have no further freedom of choice; the fifth number must be 37. Degrees of freedom (df) in our present example is one less than the number of observations. That is, df = ) 1 ( n Sometimes the formula for estimating a population parameter contains a value which is itself an estimate. Thus, to estimate a standard deviation, knowledge of the mean is required. Because the value of the mean is itself estimated from a sample, this ‘costs’ a degree of freedom (this cost is Elliott’s ‘tax’ referred to in Section 1.4). The degrees of freedom are not always ) 1 ( n ; they depend on the particular estimation in hand. We explain the rule for deciding the degrees of freedom for each technique as it arises. Degrees of freedom are symbolized by v (nu). 1.8The coefficient of variation (CV) The standard deviation is a measure of the degree of variability in a sample which estimates the corresponding parameter of a population. However, it is of limited value for comparing the variability of samples whose means are appreciably different. The standard deviation of the mass of a population of elephants is hundreds of kilograms whilst that of a population of harvest mice is a few grams. When comparing 11
variability in samples from populations with different means, the coefficient of variation (CV) is used. This is the ratio of the standard deviation to the mean, usually expressed as a percentage by multiplying by 100.
- Fall '18
- F. TAILOKA
- Standard Deviation, Mean, 1.8 g, 0.54 g, 1.7 degrees, 0.26 g