Standard Deviation
Simple Example of Calculating Standard Deviation
Let's say we wanted to calculate the standard deviation for the amounts of gold coins pirates on a
pirate ship have.
There are 100 pirates on the ship. In statistical terms this means we have a population of 100. If
we know the amount of gold coins each of the 100 pirates have, we use the standard deviation
equation for an entire population:
What if we don't know the amount of gold coins each of the 100 pirates have? For example, we
only had enough time to ask 5 pirates how many gold coins they have. In statistical terms this
means we have a sample size of 5 and in this case we use the standard deviation equation for a
sample of a population:
The rest of this example will be done in the case where we have a sample size of 5 pirates,
therefore we will be using the standard deviation equation for a sample of a population.
Here are the amounts of gold coins the 5 pirates have:
4, 2, 5, 8, 6.
Now, let's calculate the standard deviation:
1. Calculate the mean:
2. Calculate
for each value in the sample:
3. Calculate
:
4. Calculate the standard deviation:
The standard deviation for the amounts of gold coins the pirates have is 2.24 gold coins.
Standard Deviation and Variance
Deviation just means how far from the normal
Standard Deviation
The Standard Deviation is a measure of how spread out numbers are.
Its symbol is σ (the greek letter sigma)
The formula is easy: it is the square root of the Variance. So now you ask, "What is the
Variance?"
Variance
The Variance is defined as:
The average of the squared differences from the Mean.
To calculate the variance follow these steps:
Work out the Mean (the simple average of the numbers)
Then for each number: subtract the Mean and square the result (the squared difference).
Then work out the average of those squared differences.
Example
You and your friends have just measured the heights of your dogs (in millimeters):
The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.
Find out the Mean, the Variance, and the Standard Deviation.
Your first step is to find the Mean:
Answer:
600 + 470 + 170 + 430 + 300
1970
=
= 394
5
5
so the mean (average) height is 394 mm. Let's plot this on the chart:
Mean =
Now, we calculate each dogs difference from the Mean:
To calculate the Variance, take each difference, square it, and then average the result:
So, the Variance is 21,704.
And the Standard Deviation is just the square root of Variance, so:
Standard Deviation: σ = √21,704 = 147.32... = 147 (to the nearest mm)
And the good thing about the Standard Deviation is that it is useful. Now we can show which
heights are within one Standard Deviation (147mm) of the Mean:
So, using the Standard Deviation we have a "standard" way of knowing what is normal, and what
is extra large or extra small.
Rottweilers are tall dogs. And Dachshunds are a bit short
But ... there is a small change with Sample Data
Our example was for a Population (the 5 dogs were the only dogs we were interested in).
But if the data is a Sample (a selection taken from a bigger Population), then the calculation
changes!
When you have "N" data values that are:
The Population: divide by N when calculating Variance (like we did)
A Sample: divide by N-1 when calculating Variance
All other calculations stay the same, including how we calculated the mean.
Example: if our 5 dogs were just a sample of a bigger population of dogs, we would divide by 4
instead of 5 like this:
Sample Variance = 108,520 / 4 = 27,130
Sample Standard Deviation = √27,130 = 164 (to the nearest mm)
Think of it as a "correction" when your data is only a sample.
1. What is the population standard deviation for the numbers: 75, 83, 96, 100, 121 and 125?
A 16.9
B 17.1
C 17.6
D 18.2
2. Ten friends scored the following marks in their end-of-year math exam:
23%, 37%, 45%, 49%, 56%, 63%, 63%, 70%, 72% and 82%
What was the standard deviation of their marks?
A 15.1%
B 15.5%
C16.9%
D18.6%
3. A booklet has 12 pages with the following numbers of words:
271, 354, 296, 301, 333, 326, 285, 298, 327, 316, 287 and 314
What is the standard deviation number of words per page?
A 22.6
B22.0
C21.9
D21.4
4. Nine friends each guessed the number of marbles in a jar.
When the answer was revealed they found they had guessed well (and one was the
winner!)
Here is how close they each got:
-9, -7, -4, -1, 0, 2, 7, 9, 12
(A negative number shows an underestimate, a positive number shows an overestimate.)
What was the standard deviation of their errors?
A 3.9
B 5.5
C 6.2
D 6.8
5. Emma rolled a die a number of times and recorded her results in a bar graph, as follows:
What was the variance?
A3.25
B2.92
C1.89
D1.80