DescriptiveStatistics

Outline of Today’s Discussion1. Central Tendency2. Dispersion3. Graphs4. Excel Practice: Computing the S.D.5. SPSS: Existing Files6. SPSS: Entering Data

Part 1

Central Tendency

The Research Cycle

Real World

ResearchRepresentation

ResearchResults

ResearchConclusions

Abstraction

Data Analysis

MethodologyGeneralization

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Central Tendency1. One of the themes in our course will be contrasting

so-called “inferential statistics” from “descriptive statistics”.

2. Inferential statistics are used to determine (“infer”) whether two populations (or conditions) are significantly different from each other.

3. By contrast, descriptive statistics are simply used to depict (“describe”) the data in a study. We’ll focus on three descriptive measures of central tendency…

Central Tendency1. The crudest measure of central tendency is the mode

- the most frequently occurring score.

2. Here are some examples: The modal number of eyes is two.

The modal number of fingers per hand is five.The modal number of years to graduate is four.

Other examples?

3. A frequency distribution can have one mode, or two modes (bi-modal), or more (multi-modal).

Central Tendency1. The next most precise measure of central tendency is

the median - the middle score. It is the 50th percentile, i.e., the point at which half of the scores are greater and half are less.

2. The median is equal to the middle value when the data set contains an odd number of items [3, 4, 5, 6, 7, 8, 8].

3. The median is equal to the the average of the two middle two values when the data set contains an even number of items.

4. Median of [3, 5, 5, 7, 8, 8] = (5+7)/2 = 6.

Central Tendency1. The median is the best measure of central

tendency when one side of a frequency distribution contains a few, extreme scores.

2. Would someone give us some examples of “skewed” distributions (ones having a few extreme scores)?

Central Tendency1. The most commonly used measure of central

tendency is the mean - the arithmetic average.

2. There are many symbols for the mean. For a population, we use , pronounced Myou. For a sample, we simply use “M”.In computations, we use “X bar”.

3. Mean = X / N (i.e., the sum of X over N).

Central Tendency1. Here’s some sample syntax for the measures of

central tendency in Excel….

2. “ =mode(a1:a9)”

3. “ =median(a1:a9)”

4. “ =average(a1:a9)”

5. Questions on measures of central tendency?

Part 2

Dispersion

Dispersion1. When describing the data (i.e., when

generating DESCRIPTIVE STATISTICS), we want to know how the scores are distributed (“dispersed”) around the center.

2. There are several measures of dispersion.

3. We’ll consider two (for now), the range, and the standard deviation.

Dispersion1. The range is a crude measure of dispersion. It is

computed as Range = Max - Min.

2. In the set of scores [2, 4, 5, 9], the range of the scores would be Max - Min = 9 - 2 = 7 units.

3. Sometimes, rather than reporting the range, researchers will simply report the Max & Min scores.

Dispersion1. Now, let’s consider the standard deviation, which is

the most commonly used measure of dispersion (i.e., the counterpart to the mean).

2. Potential Pop Quiz Question: What information does the standard deviation provide, in your own words, (no equations here).

3. To compute the standard deviation, we first need to compute a few important quantities. One of these is called the Sum of Squares or SS…

Dispersion

1. The first step is to get the deviation of each score from the mean. Here Mean = 8.

2. Then, we square the deviations, and sum them…

Dispersion

1. So, the SS is the sum of the squared deviations from the mean.

2. In this case SS = 44.

Dispersion

1. Here are two ways to look at the SS.

2. Often, we have a definitional formula, and an equivalent computational formula.

Dispersion1. The next step in computing the standard

deviation is to determine the “variance”.

2. Variance - the average squared deviation from the mean (so, the variance, itself, is a mean).

3. We can compute the variance of either a population (i.e., every member of a group) or a sample (i.e., just a subset of a group)…

Dispersion

1. Note: The only difference is whether we divide by N (for population), or n-1 (for sample).

2. Assuming SS is constant, which formula will generate a larger variance?

Sigma “s”

Dispersion1. Finally, to get from squared units to “regular”

units, we need to take the square root of the variance.

2. The standard deviation is the square root of the variance. (Say it with me.)

3. This is true whether we’re talking about the SD of a population, or a sample…

Dispersion

The standard deviation is the square root of the variance!!

Sigma “s”

Dispersion

Another way to express the Standard Deviation(we’ve substituted the SS formula in the numerator)

Dispersion1. Phew!! That was a lot!!! Let’s review the

concepts in dispersion.

2. We have focused on two measures of dispersion; Range and Standard Deviation (SD);

3. Range is simple. It’s the Max - Min.

Dispersion1. The standard deviation indicates, approximately,

how far, on average, a score departs from the mean.

2. The standard deviation depends on some important quantities - the SS and the Variance.

3. The formula for the population variance (based on N) is slightly different than that for the sample variance (based on N-1).

4. The standard deviation is the square root of the variance!

Part 3

Graphs

Graphs1. Let’s learn some terminology about graphs.

2. The x-axis (the horizontal axis) is called the abscissa.

3. The y-axis (the vertical axis) is called the ordinate.

4. By convention, which axis contains the IV, and which axis contains the DV?

5. When describing a graph verbally, we typically state “..in this graph, (DV) is plotted as a function of (IV).”

Graphs

Describe how the variables are plotted in this graph.How many “levels” does the IV have, and what are they?

Graphs1. Sometimes an experiment has more than one IV. This

is called a factorial experiment.

2. Graphs from factorial experiments typically plot one of the IV’s on the abscissa, and the other IV by using different symbols (sometimes called parameters) in the legend.

3. Verbally, we state, “.. in this graph, (DV) is plotted as a function of IV#1, with IV#2 as parameters.”

Graphs

Describe how the variables are plotted in this graph.How many IV’s, and how many levels of each?

Graphs

Describe how the variables are plotted in this graph.How many IV’s, and how many levels of each?

Graphs

Here are the data from the preceding graph.Graphs are interpreted more quickly than tables are.

Graphs1. Another important point about graphs is that

the ordinate should start at zero.

2. If not, the graph will lose proportionality, and become very misleading.

3. Small differences could appear huge…

Graphs

Which graph is “messed up”?

Part 3

Excel Practice:Computing the S.D.

Part 4

SPSS Practice:Existing Files

SPSS: Existing Files1. SPSS data files have an extension of “.sav”

2. In variable view, each row corresponds to a new variable. The columns indicate how the variable is ‘defined’ to SPSS.

3. In data view, each row corresponds to a different participant (called a ‘case’ in SPSS). Each column pertains to a different variable.

4. Note that variables are in rows in variable view, but in columns in data view. (“It rotates by 90 degrees.”)

Part 5

SPSS Practice:Entering Data

SPSS: Entering Our Own Data1. When entering your data in SPSS, always begin in the

“VARIABLE VIEW”.

2. Each variable is in a separate row in variable view.

3. Here’s a good habit…make your first variable one that identifies the participant in some way.

4. In the type column, you should use ‘numeric’ unless you want to directly enter words on the data view. If so, you should change the type column to ‘string’.

5. In the measure column, you have three choices; Nominal, Ordinal, or Scale (which is interval and ratio).

SPSS: Entering Our Own Data1. Value labels, can facilitate data coding! These are

defined in variable view.

2. Example on Religion: 1=Buddhist; 2=Hindu;3=Islamic; 4=Jewish; 5=Christian;

3. In data view, under the view menu the value labels can be ‘turned on’ or ‘off’!