Variability

Remember, when we started at the beginning of the week, we were talking about the need to communicate two essential pieces of information about our sample. What is typical or commonly seen and the second piece of information how much difference is commonly seen.

Figure 5.1

Figure 5.2

Variability

We can measure variability by 3 different measures Range—simpy difference between the highest and

lowest scores This can show distance; whereas mean only shows

location Shows a quick check on the spread of scores Can also show errors in data; if min or max is an

implausable number

Variability

Two limitations of the Range It is not a stable measure because it is solely based on

the extreme scores in a distribution which can vary widely

Secondly it says nothing about what happens in between these two extremes

What is needed is a measure that is sensitive to every score in the distribution

Variability

If we calculate a deviation score (X - mean) this shows us the distance a score is from the mean, but remember if we sum all the deviation scores we get zero.

However, if we perform a mathematical procedure where we square those difference scores then the negative numbers become positive and we eliminate this problem

Variability

Two measures of variability that is based on the squared deviation scores Variance denoted by s2 = Σ ( xi - mean )2 / ( n - 1 )

The variance is the “mean” of the squared deviation scores

It is important to introduce a new term SS = sum of squares which is the numerator of the equation for the variance

Standard deviaton denoted by s = sqrt [ s2 ] = sqrt [ Σ ( xi - mean )2 / ( n - 1 ) ]

Table 5.1

Communicating Commonness

Another way of thinking about variance is to think of it as saying how infrequent or unusual a datapoint is in a sample.

So how unusual is this?

Variance

In addition to measures of central tendency it is critical that we revisit variance. Variance is the amount of spread within a sample. There are several common measures of variance that are typically used or reported.

Range (lowest value to highest value)

Predominance of Variance and Standard Deviation

The vast majority of the time we like to use variance and standard deviation as estimates of ranges.

This is in part due to the reduced (but still sensitive) influence of extreme scores on standard deviations.

Commonness in the Normal Distribution

One way of thinking about how common a score is what percentage of people fall in that range.

68% of people fall with in 1 SD of the mean95% fall within 2 SDIf you think back to the box graphs we can

think about the 25% and 75% as being a similar but smaller measure of commonness

Figure 5.3

Comparing Typicality and Commonness

So, one of the questions we are often asked in research is does a member of Group A typically differ from Group B. In order to answer this question we need to know what is typical (mean) for Group A and what is commonness for Group A. We need to have the same information for Group B.

Figure 5.4

Effect Size

An effect size is a critical concept to understand. Essentially it is looking to see how large the magnitude of a difference between two groups can be. We already are familiar with the two essential ingredients for estimating effect size. The mean and standard deviation.

To calculate: (Meana - Meanb)/((SSa + SSb)/(na + nb)).

Table 5.3

Variability

Look at example for effect size on page 75. This concept is very important and we will re-visit it several times throughout the semester.

Cohen suggest that we interpret effect size as .80 and above = large effect size .50 = medium effect size .20 = small effect size