Whether multiplying the same number to each value in the data set will affect its variance, please can you gave me an example.
Thank you for your help.
Suppose the dataset is:
x(i)=1,3,5,7,9
sum=∑x=25
mean=μ=25/5=5
x(i)^2=1,9,25,49,81
sum of squares=∑x^2=165
Var(x)=(165-5^2)/5=28
If all the numbers in the dataset are multiplied by 2:
x(i)=2,6,10,14,18
sum=∑x=50
mean=μ=50/5=10
x(i)^2=4,36,100,196,324
sum of squares=∑x^2=660
Var(x)=(660-10^2)/5=112 (=4*28)
What can you say about the effect on variance of multiplying the dataset by a constant?
Yes, multiplying the same number to each value in a data set will affect its variance. Here's an example to help explain it:
Let's consider a data set with the following values: {2, 4, 6, 8, 10}. The mean of this data set is calculated by summing all the values and dividing by the total number of values, which in this case is (2+4+6+8+10)/5 = 6.
To find the variance, we need to calculate the squared difference between each value and the mean, sum up these squared differences, and divide by the total number of values. The formula for variance is: variance = Σ((x - μ)²)/n, where Σ represents sum, x is each individual value, μ is the mean, and n is the total number of values.
So, for our initial data set, the squared differences from the mean are: (2-6)² = 16, (4-6)² = 4, (6-6)² = 0, (8-6)² = 4, (10-6)² = 16.
Adding up these squared differences: 16 + 4 + 0 + 4 + 16 = 40.
Dividing by the total number of values, we get 40/5 = 8. So, the variance of the initial data set is 8.
Now, let's multiply each value in the data set by 2: {4, 8, 12, 16, 20}. The mean of this new data set is (4+8+12+16+20)/5 = 12.
Calculating the squared differences from the new mean: (4-12)² = 64, (8-12)² = 16, (12-12)² = 0, (16-12)² = 16, (20-12)² = 64.
Adding up these squared differences: 64 + 16 + 0 + 16 + 64 = 160.
Dividing by the total number of values, we get 160/5 = 32. So, the variance of the new data set is 32.
As you can see, multiplying the same number to each value in the data set affects the spread of the data and, therefore, its variance. In this example, the variance increased from 8 to 32 when we multiplied each value by 2.