Answer: The addition of the new point alters the previous standard deviation greatly
Step-by-step explanation:
Let the initial five points be : 2 3 4 5 and 6. In order to calculate the standard deviation for this data, we will need to calculate the mean first.
Mean = summation of scores/number of scores.
The mean is therefore: (2+3+4+5+6)/5 = 20/5 = 4.
We'll also need the sum of the squares of the deviations of the mean from all the scores.
Since mean = 4, deviation of the mean from the score "2" = score(2) - mean (4)
For score 3, it is -1
For 4, it's 0
For 5 it's 1
For 6 it's 2.
The squares for -2, -1, 0, 1, and 2 respectively will be 4, 1 , 0, 1, 4. Summing them up we have 10 i.e (4+1+0+1+4=10).
Calculating the standard deviation, we apply the formula:
√(summation of (x - deviation of mean)^2)/N
Where N means the number of scores.
The standard deviation = √(10/5) = 1.4142
If we add another score or point that is far away from the original points, say 40, what happens to the standard deviation. Let's calculate to find out.
i.e we now have scores: 2, 3, 4, 5, 6 and 40
We calculate by undergoing same steps.
Firstly mean. The new mean = (2+3+4+5+6+40)/6 = 60/6 = 10.
The mean deviations for the scores : 2, 3, 4, 5, 6 and 40 are -8, -7, -6, -5, -4 and 30 respectively. The squares of these deviations are also 64, 49, 36, 25, 16 and 900 respectively as well. Their sum will then be 1090. i.e. (64+49+36+25+16+900 = 1090).
The new standard deviation is then=
√(1090/6)
= √181.67
= 13.478.
It's clear that the addition of a point that's far away from the original points greatly alters the size of the standard deviation as seen /witnessed in this particular instance where the standard deviation rises from 1.412 to 13.478
