Basic Statistical Concepts

Psych 231: Research Methods in Psychology

Turn in Journal summary #2 in class on Wednesday (moved from turning in last week in labs)

There are three main measures of center Mean (M): the arithmetic average

• Add up all of the scores and divide by the total number

• Most used measure of center Median (Mdn): the middle score in terms of location• The score that cuts off the top 50% of the from the bottom 50%

• Good for skewed distributions (e.g. net worth) Mode: the most frequent score

• Good for nominal scales (e.g. eye color)• A must for multi-modal distributions

Properties of distributions: Center

The Mean

The most commonly used measure of center The arithmetic average

Computing the mean

€

μ =∑X

N

– The formula for the population mean is (a parameter):

– The formula for the sample mean is (a statistic):

€

X =∑X

n

Add up all of the X’s

Divide by the total number in the population

Divide by the total number in the sample

Spread (Variability)

How similar are the scores? Range: the maximum value - minimum value

• Only takes two scores from the distribution into account

• Influenced by extreme values (outliers) Standard deviation (SD): (essentially) the average amount that the scores in the distribution deviate from the mean• Takes all of the scores into account• Also influenced by extreme values (but not as much as the range)

Variance: standard deviation squared

Variability

Low variabilityThe scores are fairly similar

High variabilityThe scores are fairly

dissimilar

mean

mean

50, 51, 48, 54, 52, 47, 45

30, 51, 38, 64, 52, 47, 65

Standard deviation

The standard deviation is the most popular and most important measure of variability. 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. • Essentially, the average of the deviations.

μ

An Example: Computing the Mean

Our population2, 4, 6, 8

€

μ =∑X

N=2 + 4 + 6 + 8

4=20

4= 5.0

1 2 3 4 5 6 7 8 9 10

μ

An Example: Computing Standard Deviation (population)

Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution.

€

μ =∑X

N=2 + 4 + 6 + 8

4=20

4= 5.0

2 - 5 = -3

1 2 3 4 5 6 7 8 9 10

μX - μ = deviation scores

-3

Our population2, 4, 6, 8

1 2 3 4 5 6 7 8 9 10

€

μ =∑X

N=2 + 4 + 6 + 8

4=20

4= 5.0

2 - 5 = -34 - 5 = -1

μX - μ = deviation scores

-1

Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution.Our population

2, 4, 6, 8

An Example: Computing Standard Deviation (population)

1 2 3 4 5 6 7 8 9 10

€

μ =∑X

N=2 + 4 + 6 + 8

4=20

4= 5.0

2 - 5 = -34 - 5 = -1

6 - 5 = +1μ

X - μ = deviation scores

1

Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution.Our population

2, 4, 6, 8

An Example: Computing Standard Deviation (population)

1 2 3 4 5 6 7 8 9 10

€

μ =∑X

N=2 + 4 + 6 + 8

4=20

4= 5.0

2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 5 = +3

μX - μ = deviation scores

3

Notice that if you add up all of the deviations they must equal 0.

Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution.Our population

2, 4, 6, 8

An Example: Computing Standard Deviation (population)

Step 2: 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 (SS).

SS = (X - μ)2

2 - 5 = -34 - 5 = -1

6 - 5 = +18 - 5 = +3

X - μ = deviation scores

= (-3)2+ (-1)2+ (+1)2+ (+3)2

= 9 + 1 + 1 + 9 = 20

An Example: Computing Standard Deviation (population)

Step 3: ComputeVariance (which is simply the average of the squared deviations (SS)) So to get the mean, we need to divide by the number of individuals in the population.

variance = 2 = SS/N

An Example: Computing Standard Deviation (population)

= 20/4 = 5.0

Step 4: Compute Standard Deviation To get this we need to take the square root of the population variance.

€

2 =X −μ( )

2∑N

standard deviation = =

An Example: Computing Standard Deviation (population)

= 5.0 = 2.24

To review: Step 1: Compute deviation scores Step 2: Compute the SS Step 3: Determine the variance

• Take the average of the squared deviations• Divide the SS by the N

Step 4: Determine the standard deviation• Take the square root of the variance

An Example: Computing Standard Deviation (population)

To review: Step 1: Compute deviation scores Step 2: Compute the SS Step 3: Determine the variance

• Take the average of the squared deviations• Divide the SS by the N-1

Step 4: Determine the standard deviation• Take the square root of the variance

An Example: Computing Standard Deviation (sample)

This is done because samples are biased to be less variable than the population. This “correction factor” will increase the sample’s SD (making it a better estimate of the population’s SD)

Relationships between variables

Example: Suppose that you notice that the more you study for an exam, the better your score typically is. This suggests that there is a relationship between study time and test performance.

We call this relationship a correlation.

Relationships between variables

Properties of a correlation Form (linear or non-linear) Direction (positive or negative) Strength (none, weak, strong, perfect)

To examine this relationship you should: Make a scatterplot Compute the Correlation Coefficient

Scatterplot

Plots one variable against the other

Useful for “seeing” the relationship Form, Direction, and Strength

Each point corresponds to a different individual

Imagine a line through the data points

Scatterplot

Hours

study

X

Exam perf.

Y

6 6

1 2

5 6

3 4

3 2

Y

X

1

2

3

4

5

6

1 2 3 4 5 6

Correlation Coefficient

A numerical description of the relationship between two variables

For relationship between two continuous variables we use Pearson’s r

It basically tells us how much our two variables vary together As X goes up, what does Y typically do

• X, Y• X, Y• X, Y

Form

Non-linearLinear

Direction

NegativePositive

• As X goes up, Y goes up

• X & Y vary in the same direction

• Positive Pearson’s r

• As X goes up, Y goes down

• X & Y vary in opposite directions

• Negative Pearson’s r

Y

X

Y

X

Strength

Zero means “no relationship”. The farther the r is from zero, the stronger the relationship

The strength of the relationship Spread around the line (note the axis scales)

Strength

r = 1.0“perfect positive corr.”

r = -1.0“perfect negative corr.”

r = 0.0“no relationship”

-1.0 0.0 +1.0

The farther from zero, the stronger the relationship

Strength

-1.0 0.0 +1.0

-.8

.5

Which relationship is stronger?

Rel A, -0.8 is stronger than +0.5

r = -0.8

Rel A

r = 0.5

Rel B

Y

X1234

56

1 2 3 4 5 6

Regression

Compute the equation for the line that best fits the data points

Y = (X)(slope) + (intercept)

2.0Change in

Y

Change in X

= slope

0.5

Y

X1234

56

1 2 3 4 5 6

Regression

Can make specific predictions about Y based on X

Y = (X)(.5) + (2.0)

X = 5

Y = ?

Y = (5)(.5) + (2.0)

Y = 2.5 + 2 = 4.5

4.5

Regression

Also need a measure of error

Y = X(.5) + (2.0) + error

Y = X(.5) + (2.0)

+ errorY

X1234

56

1 2 3 4 5 6

Y

X1234

56

1 2 3 4 5 6

• Same line, but different relationships (strength difference)

Cautions with correlation & regression

Don’t make causal claims Don’t extrapolate Extreme scores (outliers) can strongly influence the calculated relationship