Please can you Use an example to explain whether multiplying the same number to each value in the data set will affect its variance, give the relationship
Sure! To understand whether multiplying the same number to each value in a data set affects its variance, let's walk through an example.
Let's consider a simple data set of numbers: {2, 4, 6}. The variance measures the dispersion or spread of the data set. It is calculated by taking the mean (average) of the squared differences between each data point and the mean.
Step 1: Calculate the mean of the original data set.
The mean of {2, 4, 6} is (2 + 4 + 6) / 3 = 4.
Step 2: Calculate the squared differences between each data point and the mean.
The squared differences from the mean are: (2 - 4)^2 = 4, (4 - 4)^2 = 0, and (6 - 4)^2 = 4.
Step 3: Calculate the variance by taking the mean of the squared differences.
The variance is (4 + 0 + 4) / 3 = 8 / 3 ≈ 2.67.
Now, let's examine the effect of multiplying the same number to each value in the data set. Suppose we multiply each value by 3, giving us a new data set: {6, 12, 18}.
Step 1: Calculate the mean of the modified data set.
The mean of {6, 12, 18} is (6 + 12 + 18) / 3 = 12.
Step 2: Calculate the squared differences between each data point and the mean.
The squared differences from the mean are: (6 - 12)^2 = 36, (12 - 12)^2 = 0, and (18 - 12)^2 = 36.
Step 3: Calculate the variance by taking the mean of the squared differences.
The variance is (36 + 0 + 36) / 3 = 72 / 3 = 24.
Comparing the original data set's variance of approximately 2.67 to the modified data set's variance of 24, we can observe that multiplying each value by the same number has increased the variance.
Thus, multiplying the same number to each value in a data set will affect its variance. In general, multiplying the data by a constant will result in multiplying the variance by that constant squared.