• range
• Quartiles
• Interquartile range
• 5-number summaries and boxplots
• variance and standard deviation

So far we've discussed two of the three characteristics used to describe distributions, now we need to discuss the remaining - variability. Notice in our distributions that not every score is the same, e.g., not everybody gets the same score on the exam. So what we need to do is describe the varied results, rougly to describe the width of the distribution.

Variability provides a quantitiative measure of the degree to which scores in a distribution are spread out or clustered together.

Consider the two distributions to the right. They have the same shape (unimodal and symmetric) and the same center. However they are different with respect to how dispersed around their centers they are. The blue distribution has a lot of scores that are very far from the center, while most of the scores in the red distribution are very near the center. This difference is a difference in variability.

In other words variablility refers to the degree of "differentness" of the scores in the distribution. High variability means that the scores differ by a lot, while low variability means that the scores are all similar ("homogeneousness").

We'll concentrate on three measures of variability, the quartiles, the interquartile range, and the standard deviation.

Another example: height and weight of baby boys

Quartiles

Think back to percentiles. 50%tile equals the point at which exactly half the distribution exists on one side and the other half on the other side.

- Considering the same logic, what does the 25%tile represent? - The 75%?
- So using the 25th, 50th, & 75%tiles we can break the distribution into 4 quarters, or quartiles

_X f % c% 24 1 1.67 100 23 1 1.67 98.33 22 1 1.67 96.67 21 2 3.33 95.0 20 2 3.33 91.67 19 2 3.33 88.33 18 3 5.0 85.0 17 3 5.0 80.0 16 3 5.0 75.0 15 3 5.0 70.0 14 4 6.67 65.0 13 5 8.33 58.33 12 5 8.33 50.0 11 4 6.67 41.67 10 3 5.0 35.0 9 3 5.0 30.0 8 3 5.0 25.0 7 3 5.0 20.0 6 2 3.33 15.0 5 2 3.33 11.67 4 2 3.33 8.33 3 1 1.67 5.0 2 1 1.67 3.33 1 1 1.67 1.67

IQR

A related measure of variability is the interquartile range (IQR).

The interquartile range is the distance between the first quartile and the third quartile. So this corresponds to the middle 50% of the scores of our distribution.

So for the above distribution

median = Q2 = 12.5 -> using interpolation (notice exactly halfway between 12 & 13)
25%tile = Q1 = 8
75%tile = Q3 = 16

So the interquartile range (IQR) = 16 - 8 = 8.0

Note that the interquartile range is often transformed into the semi-interquartile range which is 0.5 of the interquartile range.

SIQR = (Q3 - Q1)
           2

So for our example the semi-interquartile range is (8.0)(0.5) = 4.0

So the interquartile range focusses on the middle half of all of the scores in the distribution. Thus it is more representative of the distribution as a whole compared to the range and extreme scores (i.e., outliers) will not influence the measure (sometimes refered to as being robust). However, this still means that 1/2 of the scores in the distribution are not represented in the measure.

5-number summary and box-plots.

The 5-number summary is a set of numbers that describe the spread of the distribution. It includes the Minimum and Maximum scores the median, and the first and third quartiles.



A boxplot is a graphic depiction of the 5 number summary. The center line represents the median, the red bars to the top and bottom are the quartiles, and the lines represent the largest and smallest points that are not considered outliers (may vary from stats package, typically something like +/- 1.5 IQRs determine the cut off for outliers). In SPSS you will get the boxplot of a single distribution in the Descriptive Stats - Explore submenu.

Standard deviation

The standard deviation is the most popular and most important measure of variability. It takes into account all of the individuals in the distribution.

In essence, the standard deviation measures how far off all of the individuals in the distribution are from a standard, where that standard is the mean of the distribution.

We will begin by discussion the standard deviation parameter, that is the standard deviation of the population. Then we will discuss the standard deviation statistic (for the sample). They are closely related descriptive statistics, but they have some important differences.

So to get a measure of the deviation we need to subtract the population mean from every individual in our distribution.

X - m = deviation score
- if the score is a value above the mean the deviation score will be positive - if the score is a value below the mean the deviation score will be negative

Example: consider the following data set: the distribution of heights (in inches)

69, 67, 72, 74, 63, 67, 64, 61, 69, 65, 70, 60, 75, 73, 63, 63, 69, 65, 64, 69, 65

mean = m = 67

S (X - m) = (69 - 67) + (67 - 67) + .... + (65 - 67) = ?
= 2+ 0 + 5 + 7 + -4 + 0 + -3 + -6 + 2 + -2 + 3 + -7 + 8 + 6 + -4 + -4 + 2 + -2 + -3 + 2 + -2
= 0

Notice that if you add up all of the deviations they should/must equal 0. Think about it at a conceptual level. What you are doing is taking one side of the distribution and making it positive, and the other side negative and adding them together. They should cancel each other out.



So what we have to do is get rid of the negative signs. We do this by squaring the deviations and then taking the square root of the sum of the squared deviations.

Sum of Squares = SS = S (X - m)² = (69 - 67) ² + (67 - 67) ² + .... + (65 - 67) ² =
SS = 4+ 0 + 25 + 49 + 16 + 0 + 9 + 36 + 4 + 4 + 9 +49 + 64 + 36 + 16 + 16 + 4 + 4 + 9 + 4 + 4
SS = 362

The equation that we just used (SS = S (X - m)²) is refered to as the definitional formula for the Sum of Squares. However, there is another way to compute the SS, refered to as the computational formula. The two equations are mathematically equivalent, however sometimes one is easier to use than the other. The advantage of the computational formula is that it works with the X values directly.

The computational formula for SS is:

SS = SX2 - (SX)2
             N

So for our example:

SS = [(69)2 + (67)2 + ..... + (69)2 + (65)2] - (69 + 67 + ... + 69 + 65)2
                                                              21

	     =  94631  -  (1407)2=  94631 - 94269  = 362
			    21

Now we have the sum of squares (SS), but to get the Population Variance which is simply the average of the squared deviations (we want the population variance not just the SS, because the SS depends on the number of individuals in the population, so we want the mean). So to get the mean, we need to divide by the number of individuals in the population.

Population variance = s² = SS/N

However the population variance isn't exactly what we want, we want the standard deviation from the mean of the population. To get this we need to take the square root of the population variance.
standard deviation = sqroot(variance) = sqroot(SS/N)

s = sqroot(s)

So for our example:
s² = 362 / 21 = 17.24
s = sqroot (17.2) = 4.15

To review:

step 1: compute the SS

- either by using definitional formula or the computational formula

step 2: determine the variance

- take the average of the squared deviations
- divide the SS by the N

step 3: determine the standard deviation

- take the square root of the variance

Now let's move onto the Standard Deviation of a Sample
- the computations are pretty much the same here
- different notation:
s = sample standard deviation
use instead of m in the computaion of SS

- need to adjust the computation to tak into account that a sample will typically be less variable than the corresponding population.



- if you have a good, representative sample, then your sample and population means should be very similar, and the overall shape of the two distributions should be similar. However, notice that the variability of the sample is smaller than the variability of the population.

- to account for this the sample variance is divided by n - 1 rather than just n

 sample variance = s2 = __SS _
                        n - 1

- and the same is true for sample standard deviation

sample standard deviation = s = sqroot(SS/(n - 1))

What we're really doing here is trying to use a sample to make estimates about the nature of the population. But since we don't know things like what is the mean of the population, we really can't measure our deviances from the population standard. So what we use is our best estimate of what the population mean is, and that is the sample mean.

So what we're doing when we subtract 1 from n is using degrees of freedom to adjust our sample deviations to make an unbiased estimation of the population values.

What are degrees of freedom? Think of it this way. You know what the sample mean is ahead of time (you've got to to figure out the deviations). So you can vary all but one item in the distribution. But the last item is fixed. There will be only one value for that item to make the mean equal what it does. So n - 1 means all the values but one can vary.

Example:

suppose that you know that the mean of your sample = 5
if your first 4 items are:
5, 4, 6, 2 then what must the final number be?
5 + 4 + 6 + 2 + X = 25
there will be only one value of X that'll make this work. X = 8

Okay, so let's do an example of computing the standard deviation of a sample

data: 1, 2, 3, 4, 4, 5, 6, 7

step 1: compute the SS

SS = S (X - )²
= (1 - 4)² + (2 - 4)² + (3 - 4)² + (4 - 4)² + (4 - 4)² + (5 - 4)² + (6 - 4)² + (7 - 4)²
= 9 + 4 + 1 + 0 + 0 + 1 + 4 + 9 = 28


-- OR --

You can still use the computational formula to get SS

SS = SX2 - (S X)2
             N

		  = (1+4+9+16+16+25+36+49) - (1+2+3+4+4+5+6+7)
                                                      8
		  = 156 - 128 = 28.0

step 2: determine the variance of the sample (remember it is a sample, so we need to take this into account)

sample variance = s2 = _SS_
                      n - 1


= 28/(8-1) = 28/7 = 4.0

step 3: determine the standard deviation of the sample
standard deviation = sqroot(SS/(n - 1))
= sqroot(28/(8 - 1)

= sqroot 4.0 = 2.0

Properties of the mean and standard deviation (Transformations)

1) Adding a constant to each score in the distribution will not change the standard deviation but will change the mean (by that same constant).

So if you add 2 to every score in the distribution, the mean changes (by 2), but the variance stays the same (notice that none of the deviations would change because you add 2 to each score and the mean changes by 2).

2) Multiplying each score by a constant causes both the mean and stardard deviation to be multiplied by the same constant.
This one is easier to think of with numbers. Suppose that your mean is 20, and that two of the individuals in your distribution are 21 and 23. If you multiply 21 and 23 by 2 you get 42 and 46, and your mean also changes by a factor of 2 and is now 40. Before your deviations were (21 - 20 = 1) & (23 - 20 = 3). But now, your deviations are (42 - 40 = 2) & (46 - 40 = 6). So your deviations are getting twice as big as well.

Comparing Measures of Variability

- Extreme scores: range is most affected, IQR is least affected
- Sample size: range tends to increase as n increases, IQR & s do not
- The range does not have stable values when you repeatedly sample from the same population, but the IQR & S are stable and tend not to fluctuate.
- With open-ended distributions, one cannot even compute the range or S, so the IQR (or SIQR) is the only option