Central Tendencies

Central Tendency and Variability

Important Points
The standard deviation is the most commonly used measure of variable with measurement data.

Things to Know:

1. The standard deviation is a landmark on the normal curve. It reaches from the mean to the point of inflection of the normal distribution. What does this mean? – See the picture below.

2. Given the mathematics of the normal curve, there are always 68.26% of the scores between the standard deviation marks.

3. So if someone tells you the mean and standard deviation you can tell the two scores that encompass a little more than 2/3s of your distribution

Let’s look at our shoe data.

If you had calculated the standard deviation of the distribution of shoe sizes and found out it was 2.0* you would know that 68.26 % of males probably have shoe sizes between 7 and 11. This 9 + 1 SD or 9 + 2

*(Hint: look back at our table with the normal depression in it and figure out the standard deviation visually)

Now, if I knew this information, I could figure how to order my shoes a little better. I could order sizes more in the center. At least, I would know not to buy a lot of size 4,5, 13 and 14. It would be better to buy a lot of 9s and a good deal of 8s and 10s and a few 7s and 11s.

Preview – in Workshop 2 – you will see how to pin down these percentages even more exactly.

Let’s apply this to other situations!

The Mean IQ on the Stanford-Binet IQ test is 100. The Standard Deviation is 16. Thus 68.26% of people have scores between 84 and 116.
The Mean of the SAT-V is 550 and the SD is 100. Thus 68.26% have scores between 450 and 650.
One college reports its SAT-V mean has 600 and its standard deviation at 50. Another reports it as a mean of 500 and a SD of 100. You know that the first school is higher and less variable than the second is.

Nuances:

When you calculate the SD deviation from a sample you divide by N –1 rather than N because sample SDs tend to be too small.
The variance is useful later in comparing groups
If you have non-normal distributions, you might have to shift techniques.

Bottom Line:

If you have a data set, you need to know what’s typical and how are all the scores spread out. If you know this, you can better understand the situation. We will see how this leads to better prediction and decision making.