STANDARD DEVIATION & Z-SCORES

Let’s start with an example

I divided the class into 2 teams, A and B. Coincidentally, the quiz average for team A is the same as team B, 81.5. So we expect a graph of their scores to be about the same, right?

Not so!

Let’s look at the scores:

Mean

Standard Deviation is a number that tells us…

How far from “typical” a certain piece of data is.

How spread out the data items are.

Calculating the Standard Deviation, σ (sigma)

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.

Step 1: Calculate the meanMean = 600 + 470 + 170 + 430 + 300 5

Mean = 1970 = 394 5

So the average height is 394 mm

Step 2: Calculate each dog’s difference from the mean.

Step 3: Square each of those results.Step 4: average the result.

We call this average the variance (σ2 )

σ2 = 2062 + 762 + (-224)2 + 362 + (-94)2

5σ2 = 108,520 = 21,704

5

Step 5: Find the square root of the variance σ2 to get the standard deviation σ So, the Variance = 21,704. And the Standard Deviation is just the

square root of Variance, so:

Standard Deviation: σ = √21,704 = 147

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.  

Rottweillers are tall dogs. And Dachsunds are…well, a bit short ... but don't tell them!

Z-scoreZ-score – also called standard score indicates how many standard deviations an observation is above or below the mean.z-score =

What is the z-score for the Rottweiler? Z-score = 1.4 σ

𝑥 𝑖−𝜇𝜎