Psychological Statistics

Psychology is a quantitative endeavor.

Operationalism demands that we specify the means of measurement of concepts we study

Psychological statistics come in two varieties:

Descriptive StatisticsInferential Statistics

Raw Data

For Example - A psychologist conducts an experiment on learning with a sample of 40 rats.

Each rat is run through a complex maze with many turns and the number of incorrect turns (Errors) is recorded. Each rat is run repeatedly through the maze until “mastery” is attained (defined operationally as three consecutive trials without an error).

Raw Data

For Example - A psychologist conducts an experiment on learning with a sample of 40 rats.

Each rat is run through a complex maze with many turns and the number of incorrect turns (Errors) is recorded. Each rat is run repeatedly through the maze until “mastery” is attained (defined operationally as three consecutive trials without an error).

The measure of principal interest is the “Number of Trials to Mastery”

Descriptive Statistics

Statistics that describe or summarize collections of data

Description can be tabular (Table), pictorial (Graph), or might involve the computation of an index (e.g., the mean)

Raw Data Frequency Distribution

15 14 22 14

20 8 13 15

12 20 12 13

9 19 16 15

16 18 17 11

18 11 14 13

7 10 15 17

11 13 14 21

12 16 9 10

14 12 21 14

Score Frequency

7 1

8 1

9 2

10 2

11 3

12 4

13 4

14 6

15 4

16 3

17 2

18 2

19 1

20 2

21 2

22 1

Measures of Central TendencyMeasures of central tendency provide us with an index of where the center of the distribution is located

Mean (X) – The arithmetic average

ΣX

NX =

Computing the Mean

15 14 22 14

20 8 13 15

12 20 12 13

9 19 16 15

16 18 17 11

18 11 14 13

7 10 15 17

11 13 14 21

12 16 9 10

14 12 21 14

ΣX = 571

N = 40

X = 14.275

Measures of Central Tendency

Median – A point in the distribution at which exactly half of the scores are higher and exactly half are lower

If the distribution has an odd number of scores, arrange them in ascending order and simply select the middle score

If the distribution has an even number of scores, arrange them in ascending order and take the average of the middle two

Mode – The most commonly incurred score

Unimodal

Bimodal

Measures of Variability

These measures indicate how dispersed or spread out the scores are from the center of the distribution

Measures of Variability

These measures indicate how dispersed or spread out the scores are from the center of the distribution

Range = Highest Score – Lowest Score

Measures of Variability

The Standard Deviation is a more sophisticated measure of variability than the range. Range can be

significantly distorted by outliers.

σ =

Normal Distributions

These measures indicate how dispersed or spread out the scores are from the center of the distribution

Virtually all characteristics studied by psychologists (as well as other scientists) are likely to be normally distributed.

Skewed Distributions

Positive Skew

Skewed Distributions

Negative Skew

Correlation

Correlation refers to a relationship that exists between pairs of measures. Knowledge of the strength and direction of the relationship allows us to predict one variable from the other with an accuracy greater than chance.

For example, you can guess someone’s weight more accurately if you know how tall they are because height and weight are positively correlated.

When two variables are positively correlated that means they tend to both move higher or both move lower at the same time.

Generally, taller people weigh more than shorted people.

When two variable are negatively correlated that means that they change in value inversely. Higher scores on one generally go with lower scores on the other.

For example, the outdoor temperature and the weight of one’s clothing are negatively correlated. The higher the temperature is, the less clothing we wear. The lower the outdoor temperature, the more clothing we wear.

Correlation

Correlations can be represented graphically by the construction of a special type of graph called a scatter-plot diagram. Pairs of

scores are obtained from each subject.

For example, suppose we wanted to look at the relationship between height and self esteem in men. Perhaps we have a

hypothesis that how tall you are effects your level of self esteem. So we collect pairs of scores from twenty male

individuals. Height, measured in inches, and Self Esteem based on a self-rating scale (where higher scores mean

higher self esteem).

Man HeightSelf

Esteem

1 68 4.1

2 71 4.6

3 62 3.8

4 75 4.4

5 58 3.2

6 60 3.1

7 67 3.8

8 68 4.1

9 71 4.3

10 69 3.7

11 68 3.5

12 67 3.2

13 63 3.7

14 62 3.3

15 60 3.4

16 63 4.0

17 65 4.1

18 67 3.8

19 63 3.4

20 61 3.6

Scatter-Plot Diagrams

Coefficient of Correlation

A statistical computation that indicates the strength and direction of an underlying correlation

Always results in a signed number in the range from -1.00 to +1.00 If the sign is positive, that indicates the underlying

relationship is a positive correlation. If the sign is negative, it indicates an underlying negative correlation. The closer the value is to “1” (either positive or negative) the stronger is the

indicated underlying relationship.

Inferential Statistics

Inferential statistics Interpret or in other words allow us to draw conclusions about the meaning of our data.

We want to be able to draw conclusions that apply universally – that is to the population at large.

Since we can’t work with all members of a population, we draw a small subset from it called a “sample”.

Extrapolating from our findings using subjects from the sample, we then draw conclusions about the overall population.

Our success depends upon the representativeness of our sample. We rely on Randomization as a selection procedure,

large Sample Sizes, and Replication (Multiple Samples) to insure representativeness.

Hypothesis Testing

The ultimate test of any hypothesis would be to subject it to an experimental test.

The experiment is a research method in which the investigator manipulates a variable (IV) under

carefully controlled conditions and observes whether any changes occur in behavior (DV) as a result.

Example of an Experiment

Population – Insomniacs (people who report difficulty in falling asleep)

Research Hypothesis – A new drug (hypnoxin) may reduce the delay of sleep onset in this population.

Null Hypothesis Testing – The Null Hypothesis is the very conservative assumption that our drug will not produce any effective relief. We start with assumption and stick

with it unless the data overwhelming convince us otherwise.

Example of an Experiment

Subjects are randomly assigned to two groups. The experimental group includes subjects that are to receive

our new drug. A control group would also be tested under identical circumstances except they don’t receive

the drug.

To protect against a “placebo effect”, all subject receive an identically appearing pill with the instructions “Here

take this it may help you sleep”. For subjects in the Experimental Group the pill contains Hypnoxin, For

subjects in the Control Group the pill contains an inert ingredient. Subjects don’t know which condition they

have been assigned to.

The number of minutes to sleep onset after taking the pill and retiring is recorded as the DV.

Subject Hynoxin Control

1 43 42

2 12 88

3 22 19

4 45 41

5 55 74

6 14 66

7 23 24

8 44 17

9 15 29

10 35 38

11 14 51

12 51 57

Mean 31.08 45.50

Delay of Sleep Onset with and without Hypnoxin

Is the difference between the two conditions sufficiently large for us to reject the Null Hypothesis? Or could the difference be due to chance alone? If the probability that this is due to chance alone is

less than .05 we consider the outcome to be statistically significant.

Science Does Not Discover Truth

Science is probabilistic.

DNA Testing from a crime scene – Conclusion is probabilistic “There is a 1 in 1 billion chance that the DNA recovered might be from someone other than

the suspect”