Statistics for the Social Sciences

Statistics for the Social SciencesPsychology 340

Spring 2005

Distributions

Statistics for the Social Sciences

Outline (for week)

• Variables: IV, DV, scales of measurement– Discuss each variable and it’s scale of measurement

• Characteristics of Distributions– Using graphs– Using numbers (center and variability)

• Descriptive statistics decision tree

• Locating scores: z-scores and other transformations


Let’s get some distributions

• On a sheet of paper (that you’ll turn in) write out these pieces of information:– Male or female– Height (in inches)– How many pairs of shoes in your closet– Typical number of servings of soda per day– Typical number of servings of water per day

• Okay, turn these in


Basic Concepts

Variable– A condition or characteristic that can have different

values Value

– A possible number or category that a score can have Score

– A particular person’s value on a variable


Basic Concepts

Kinds of Variables– A condition or characteristic that can have different

values– Experiment:

– Independent - manipulated by experimenter– Dependent - measured by experimenter

– Observational:– Explanatory - observed variable to do the explaining– Response - variable to be predicted


Measurement

• Properties of our measurement?– Units of measurement - whether the measurement has a

minimum sized unit or not– Levels (Scales) of measurement - the correspondence

between the numbers representing the properties that we’re measuring


Units of Measurement

• Continuous variables– Variables can take any number and can be infinitely

broken down into smaller and smaller units– E.g., For lunch I can have 2, 3, or 2.5 cookies

• Discrete variables– Broken into a finite number of discrete categories that can’t be broken down– E.g., In my family I can have

1 kidor 2 kids, but not 2.5


Levels (scales) of measurement

• Nominal Scale: Consists of a set of categories that have different names. – Measurements on a nominal scale label and categorize

observations, but do not make any quantitative distinctions between observations.

– Example:• Eye color:

blue, green, brown, hazel


Levels of measurement

• Ordinal Scale: Consists of a set of categories that are organized in an ordered sequence. – Measurements on an ordinal scale rank observations in

terms of size or magnitude.– Example:

• T-shirt size: Small, Med, Lrg, XL, XXL



• Interval Scale: Consists of ordered categories where all of the categories are intervals of exactly the same size. – With an interval scale, equal differences between

numbers on the scale reflect equal differences in magnitude.

– Ratios of magnitudes are not meaningful.– Example:

• Fahrenheit temperature scale20º40º

“Not Twice as hot”



• Ratio scale: An interval scale with the additional feature of an absolute zero point. – With a ratio scale, ratios of numbers DO reflect ratios of

magnitude.


Our variables

• Descrete or continuous?• What level of measurement?

– Male or female– Height (in inches)– How many pairs of shoes in your closet– Typical number of servings of soda per day– Typical number of servings of water per day

• Now let’s consider how we would describe one of these variables


Distributions

• The distribution of a variable is a description of all of the tokens of the variable within in sample (or population if you’ve got the data)– A picture of the distribution is usually helpful

• Gives a good sense of the properties of the distribution

– Many different ways to display distribution• Frequency distribution table• Graphs


Steps for Making a Frequency Table(do this for class soda drinking variable)

• Make a list down the page of each possible value, from highest to lowest

X f % c% 12 11 10 9 8 7 6 5 4 3 2

The values of the variable


Steps for Making a Frequency Table

• Go one by one through the scores, making a mark for each next to its value on the list, count up how frequently each value appears and include this in the table

X f % c% 12 11 10 9 8 7 6 5 4 3 2


The number of tokens of each variable



• Figure the percentage (or proportion) of scores for each value

X f % c% 12 11 10 9 8 7 6 5 4 3 2



N=total% = (f/N)*100

The percentage of tokens at each value



• Figure the cumulative percentage (or proportion) of scores for each value

X f % c% 12 11 10 9 8 7 6 5 4 3 2



N=total% = (f/N)*100

The percentage of tokens at each value

Cumulative percentage


Grouped Frequency Table(do this for class height variable)

A frequency table that uses intervals (range of values) instead of single values


Frequency Graphs

Histogram Plot the

different values against the frequency of each value


Frequency Graphs

Histogram (create one for class height) Step 1: make a frequency distribution table

(may use grouped frequency tables) Step 2: put the values along the bottom, left to

right, lowest to highest Step 3: make a scale of frequencies along left

edge Step 4: make a bar above each value with a

height for the frequency of that value


Frequency Graphs

Frequency polygon - essentially the same, put uses lines instead of bars


Properties of distributions

• Distributions are typically summarized with three features

• Shape• Center• Variability (Spread)


Shapes of Frequency Distributions

Unimodal, bimodal, and rectangular


Shapes of Frequency Distributions

Symmetrical and skewed distributions

Normal and kurtotic distributions


Displaying two variables

Bar graphs Can be used in a number of ways (including

displaying one or more variables) Best used for categorical variables

Scatterplots Best used for continuous variables


Bar graphs

• Plot a bar graph of men and women in the class

• Plot a bar graph of shoes in closet crossed with men and women– What should we plot? (and why?)

• Total number of shoes for each group?• Average number of shoes for each group?


Scatterplot

• Plot a scatterplot of soda and bottled water drinking– Useful for seeing the relationship between the

variables


Next time

• In addition to using tables and graphs to describe distributions, we also can provide numerical summaries

Documents

Statistics for the Social Sciences