Concept Sheet 2
Descriptive Statistics

After we have collected a group of data we then have to being describing it. To begin we are interested in two broad characterizations of data: measures of central tendency or location and measures of dispersion.

Central Tendency or Location

Measures of central tendency attempt to get a handle on the "middle" or representative observation. There are three different measures of central tendency. Even for one data set, the three measures can give different answers, because each measure looks at different aspects of "middle-ness."

For instance, let's say we collected information on the scores of a recent exam and found the scores to be: 75, 83, 98, 43, 92, 83, 85, 90, and 87. What is the "middle" score? Let's use the three measures of central tendency developed iin class to find the answer.

Mode --is the value with the maximum frequency in the data set, or the value that occurs most often.

In our example the mode is 83 since it occurred twice.

• One of the great advantages of the mode is that it can be used if the data collected are of nominal type, e.g. if the data set was each student's religion, rather than test scores (an interval measure). Also, there is no calculation involved.
• The potential drawbacks are that the mode ignores the rank order of the observations and does not take into account that some of the observations may be very far apart. In addition, there can be more than one mode in a particular data set.

Median --is a positional rank, or the value having the same number of observations above and below. This requires that the observations are ordinally measured.

Thus for our exam score example, we first must order the data: 43, 75, 83, 83, 85, 87, 90, 92, 98. The median here is 85, because there are 4 observations above and below the score 85. Had there been an even number of observations in the data set, we simple would have averaged the middle two to come up with the median (For example if another student took the exam and got a 93, we would have averaged 85 and 87 to come up with a median of 86.)

• The median has the advantage of not being influenced strongly by outliers (observations at extreme ends). For example, adding a score of 13 to the data set does not greatly impact the median.
• Again, as with the mode, the median does not take into account the size of the intervals between the observations therefore some information will be lost.

Mean , --is an arithmetic measure defined as the sum of all observations divided by the total number of observations.

In our original data set the mean is 81.8 (or approximately 82).

• As we will see throughout this course, the mean has wide statistical applications that will make it more attractive than either the mode or the median.
• However, the major drawback to using the mean that we must always keep in mind is that it is the most sensitive measure of central tendency to outliers. For example if we dropped the outlier in our original group of data (the score of 43), the mean would change to 86.6 (almost 87) which is significantly different from 82.

Now that we have a sense of the "middle" or typical value of the interpretation as above, we now want to get a sense of how the data vary around the "middle" that we just found. These measures used to describe the data set are called measures of dispersions.

Measures of Dispersion

We will break these measures up into two categories depending on how the data are measured.

I. For Discrete/Nominal or Ordinal Variables:
These statistics measure variation in a sample by comparing the cases (i.e., scores or observations) to one another.

Variation Ratio (V)--is the ratio of NON-modal observations to the total number of observations (or is one minus the percent of observations in the modal category).

• You can only use V for uni-modal or uniform (every category has an equal frequency) distributions.
• This measure unfortunately ignores information such as how many categories there are or if all other observations are in one (or for that matter a thousand) other category.

Minimum value for V: 0, where all cases are modal (a constant!)
Maximum value for V: 1, where no cases are modal.

Index of Diversity (D)--indicates the likelihood that two observations drawn at random from the sample are from different categories of the variable. The larger the number for D, the greater the dispersion.

where:

k = number of categories of the variable
p = proportion of cases in a given category

The problem with D is that you can only compare scores for D when the number of categories are held constant.

Minimum value for D: 0 (when all cases fall into a single category)
Maximum value for D: (K-1)/K (a function of the number of categories of the variable)

Index of Qualitative Variation (IQV)--standardized version of D. Same interpretation as D, but IQV creates a maximum value of 1, regardless of the number of categories of the variable.

Improvement over D is that we can compare IQV scores across samples with different numbers of categories (K) because it is now standardized.

II. For Continuous/Interval Variables:
With the exception of the range, these statistics measure variation in the sample by comparing the scores to the mean.

Range--difference between extreme values of a distribution (high value minus low value).

R = Highest value - Lowest value.

• The Range doesn't say anything about how many observations are in the extreme categories, hence it is really sensitive to outliers, a drwaback to its use. It also "ignores" information, much like the measures of central tendency.
  

Minimum value for the range: 0 (where all observations are on the same value)
Maximum value for the range: the distance from the lowest to the highest possible measurement

Average Deviation--sum of deviation of scores (cases or observations) about the mean, divided by the total number of scores.

• The only drawback to this measure is it does not have the desirable statistical properties that the standard deviation has (see below).

Variance--the sum of the squared deviation of the scores about the mean, divided by the total number of scores minus 1. (Note: we divide by (n-1) because, otherwise, the sample variance will be a biased estimator of the population variance. We will talk more later in the semester about what this means.)

• A drawback to the variance is that it yields a measure of dispersion in terms of squared units of the variable (e.g., squared points on the exam, or squared years of schooling, etc.). This is not easily interpretable. We can take the square root of the variance to convert the measure of dispersion back into the appropriate units of the variable which makes the number make sense so that we can interpret it.

Standard Deviation s, --the square root of the variance. Think of it as the "average" or "typical" amount of variation around the mean. An "average" observation is one standard deviation away from the mean.

Minimum value for s.d.: 0 (where there is no variation from the mean at all; all cases are on the mean)
Maximum value for s.d.: approximately half of the total possible range of measurement (for example, on a 100 point scale, max. s.d. is roughly 50)

Chebychev's Inequality Theorem:

For ANY distribution, at least (1-1/k²)*100% of the observations lie within k standard deviations of the mean.

At least 75% lie within 2 standard deviations

At least 89% lie within 3 standard deviations.

For Normally Distributed data:

About 68% lie within ONE standard deviation

About 95% lie within 2

About 99% lie within 3.

Posted February 18, 1998
Copyright Chris Fastnow