Variance, Standard deviation Exercises: 1. What does variance measure? 2. How do we compute a variance? 3. What is the difference between variance and standard deviation? 4. What is the meaning of the variance when it is negative? 5. If I add 2 to all my observations, how variance and mean will vary? 6. If I multiply the result of my observations by 3, how variance and mean will vary? 7. What is the impact of one or few values different from the others on the variance? 1
We are not able to compute variance mentally. The variance indicates the variability of a list of values. It is an average distance from the mean on the observations we have. The more different from each other our data are, the greater is the variance. Variability is a fundamental basis for statistics, if there is no variability in the phenomenon, we usually do not need statistics, if we want help from statistics we will have to measure more than one to assess variability of the phenomenon. The variance use the distance of our values from their mean. If the values are grouped near to the mean the variance will be little. Usually the variance is not accompanied with the measure scale, if it would be the case it would be the square of the unit of measure. The standard deviation when we see its formula seems more complicated than the variance (there is a square root); but it is practically easier to understand. It shows how far are the values from the mean on average in the same scale as the measure (meters, number of seeds, weight ) How do we compute a variance? Variance Case 1 Case 2 Case 3 Case 4 Case 5 value 1 10 8.5 9 9-10 value 2 10 9.5 10 9 0 value 3 10 10.5 10 11 20 value 4 10 11.5 11 11 30 mean 10 10 10 10 10 variance 0.00 1.67 0.67 1.33 333.33 used std deviation 0.00 1.29 0.82 1.15 18.26 used varp 0.00 1.25 0.50 1.00 250.00 std dev P 0.00 1.12 0.71 1.00 15.81 measure from mean value 1 0.00 1.50 1.00 1.00 20.00 measure from mean value 2 0.00 0.50 0.00 1.00 10.00 measure from mean value 3 0.00 0.50 0.00 1.00 10.00 measure from mean value 4 0.00 1.50 1.00 1.00 20.00 mean 0 1 0.5 1 15 would have been nice difference from mean value 1 0-1.5-1 -1-20 difference from mean value 2 0-0.5 0-1 -10 difference from mean value 3 0 0.5 0 1 10 difference from mean value 4 0 1.5 1 1 20 mean 0 0 0 0 0 not appropriate 2
To quantify distance from the mean we must sum distances in the common sense, which are always positive, as if we measure a distance with a wooden rule. If we use the difference between the values and the mean we would obtain a value which would indicate the fact that values are rather lower or rather bigger the mean. (see not appropriate of above table) The easier way would have been to use the average distance from the mean. This is the value, which is related to our common sense, which is equivalent to use a meter and measure distances between points (see would have been nice from above table) This approach was proposed among others to describe variability, but was not kept by mathematicians, it would have simplified our understanding if it had been kept. From the Web: A demonstration of how is computed a variance http://ouvaxa.cats.ohiou.edu/~wallace/class/standrd.html Variance case 1 case 2 Case 3 Case 4 Case 5 value 1 10 9 8 4-10 value 2 10 10 10 10 10 value 3 10 11 12 16 30 mean 10 10 10 10 10 variance 0 1 4 36 400 std deviation 0 1 2 6 20 the 3 values the average the average the average the average are equal to distance to distance to distance to distance to the mean the the mean is 1 the mean is 2 the mean is 6 the mean is distance from 20 them to the mean is 0 NB: the values had been chosen for the purpose of the simplification of the understanding 3
How does the scale of the values affects variance and mean? Variance 2 3 Values add multiply 1 3 3 2 4 6 3 5 9 4 6 12 5 7 15 6 8 18 7 9 21 8 10 24 9 11 27 10 12 30 sum 55 75 165 mean 5.5 7.5 16.5 Variance 9.17 82.50 variance P 8.25 8.25 74.25 ratio mean 1.36 3 ratio variance 1 9 90 80 70 60 50 40 30 20 10 0 Values add multiply mean variance std deviation 3.03 3.03 9.08 If we add a constant to values, the mean will increase of this constant. If we multiply our values by a constant, the mean will be multiplied by this constant. If we add a constant to values, the dispersion of the values from the mean is not changed, so the variance is not affected and remains the same. If we multiply our values by a constant, the standard deviation is multiplied by this constant, the variance is multiplied by the square of this constant Example about salaries: Not everyone have the same salary in our laboratory. We can compute a mean salary for the laboratory and a variance of the salary in the laboratory. If every one get a special bonus of 10 000 FF, each salary will be increased by 10 000 FF, as well as the mean salary. The variance will not change, the disparities between persons are not affected if every one has the same bonus. If we change the currency from FF to Euros, we still have exactly the same amount of money, but both mean and variance are different. This shows the unit of measure has an influence on mean and variance, we can not compare them for objects having different measure units. Usually in germination, purity, the units used are always the same; but in pathology tests for instance if we compare methods the units of measurements might be different, be aware. 4
From the Web: Put data on a histogram and see how mean and standard deviation are affected. http://www.ruf.rice.edu/~lane/stat_sim/descriptive/index.html look at effect of sample size on mean and variance http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html 5