Fall Semester 2002

A die (plural: dice) is a small cube marked on each face with from one to six spots and used usually in pairs in various games and in gambling by being shaken and thrown to come to rest at random on a flat surface. Let's visualize that, if you will:

E = { 1, 2, 3, 4, 5, 6 }

E = { e₁, e₂, e₃, e₄, e₅, e₆ }

E = { e₁, e₂, ..., e_m }

Note that the experiment involves throwing just one die.

The collection of all these events is called the event space for the experiment.

Describe the event space when we throw two dice (it's a set of ordered pairs).

Describe the event space when we throw two dice and are interested in the sum of their points.

If the die is unbiased, when we perform an experiment, the likelihood that we get one of the six events is the same for each event.

Assume that we do n > 0 experiments.

Let f (e_i) be the frequency of occurrence of event e_i in the n experiments.

The experimental probability of event e_i in n experiments, is defined as

The first part of the lab tomorrow will be probability related.

Next, let's review some of the things we did last time.

2. Central Statistics

Let E be a number-based event space and let

X = (x₁, x₂, ..., x_n)

The mean (also called average or expected value) of X is defined as:

• the experimental probability of event e as P(e_i) and if
• the event space E has m events,

This expression links the theory of probability to the statistical notion of mean.

The mean is a statistic of so called central tendency.

In fact, one can prove that:

If we take the sum over the differences between each one of the observed events and the mean we obtain a value of 0 (zero). The differences cancel each other out. Hence the mean is the most central value (from this point of view, almost a center of mass) that characterizes the observed events.

Let's summarize here the properties of the mean:

• The mean is sensitive to the exact value of all the scores in the distribution.

• The sum of the deviations about the mean equals zero.

• The mean is very sensitive to extreme scores.

• The sum of squared deviations about the mean is a minimum.

This last property is very important and is used in many areas of statistics, particularly in regression. Elaborated a little more fully, this property states that although the sum of the squared deviations about the mean does not usually equal zero, it is smaller than if squared deviations were taken about any other value.

You have tested this property in Lab Two or Three, and in class last Thursday.

There are other measures of central tendency, such as the median and the mode.

Let's review them.

They both rely on the idea of sorting.

Let

X = (x₁, x₂, ..., x_n)

Since the x_j's are numbers we can sort the list X in ascending (increasing) order.

Assume that the result is the list

(y₁, y₂, ..., y_n)

• the number of x_i's less than or equal to the median is equal to
• the number of x_i's greater than or equal to the median.

The third and last measure of central tendency that we shall discuss is the mode.

The mode is defined as the most frequent score in the distribution. (When all scores in the distribution have the same frequency, it is customary to say that the distribution has no mode).

Clearly, this is the easiest of the three measures to determine. The mode is found by inspection of the scores; there isn't any calculation necessary.

Usually distributions are unimodal; that is, they have only one mode. However, it is possible for a distribution to have many modes. When a distribution has two modes, the distribution is called biimodal. In general, with more than two modes a distribution is called multimodal.

Measures of Central Tendency and Symmetry.

If the distribution is unimodal and symmetrical, then the mean, median and mode will all be equal. When the distribution is skewed, the mean and median will not be equal. Since the mean is most affected by extreme scores, it will have a value closer to the extreme scores than will the median. Thus, with a negatively skewed distribution, the mean will be lower than the median. With a positively skewed curve, the mean will be larger than the median.

Here's a picture that illustrates the point:

3. Statistics of Dispersion

Measures of Variability

Variability has to do with how far the scores (or values obtained, measured in the experiments) are spread apart. Whereas measures of central tendency are a quantification of the average value of the distribution, measures of variability quantify the extent of dispersion. There are three measures of variability we will be looking at:

• the range,
• the variance, and
• the standard deviation

The range is easy to calculate but gives us only a relatively crude measure of dispersion, because the range really only measures the spread of the two extreme scores and not the spread of any of the scores in between. (By scores we mean, as usual, observed events, or measurements).

Let

X = (x₁, ..., x_n)

The variance of the list X is defined as follows:

The standard deviation of a list of experiments X is defined as follows:

The standard deviation has many important characteristics:

• it gives us a measure of dispersion relative to the mean

  This differs from the range, which tells us directly the spread of the two most extreme scores.

• it is sensitive to each score in the distribution

  If a score is moved closer to the mean, then the standard deviation will become smaller. Conversely, if a score shifts away from the mean, then the standard deviation will increase.

• it is stable with regard to sampling fluctuations (just like the mean)

  If samples were taken repeatedly from populations of the type usually encountered in the behavioral sciences, the standard deviation of the samples would vary much less from sample to sample than the range.

4. The Normal Curve

The normal curve is a theoretical distribution of population scores. It is a bell shaped curve which, like most other curves, has an equation that describes it

5. The Normal Curve and Standard Scores

5.1 The Normal Distribution

A broad range of observed phenomena in nature and in society is approximately normally distributed. For example, the distributions of variables such as

• height,
• weight,
• blood pressure,
• intelligence,
• achievement

Take a look at homework two now.

5.2 Shape of the Normal Distribution

This is only an approximate rendering, still meaningful.

5.3 Area Contained Under The Normal Curve

Appears indicated in the diagram above.

Please check these numbers in lab.

5.4 Standard Scores (z-Scores)

A z score is a transformed score that designates how many standard deviation units the corresponding raw score is above or below the mean.

Important use of z scores:

to compare scores that are not otherwise directly comparable

Take a look at and think about the new homework again now.

5.5 Characteristics of z-Scores

1. The z-scores have the same shape as the set of raw scores. (Transforming the raw scores into their corresponding z scores does not change the shape of the distribution. Not do the scores change their relative positions. All that is changed are the score values).

2. The mean of the z scores always equals zero. This follows from the observation that the scores located at the mean of the raw scores will also be at the mean of the z scores.

In lab we will also look at the following transformations:

• start from a raw score in a normal distribution, calculate percentile rank

• start from a percentile rank, calculate the raw score