Quantitative variables are ones for which we can calculate averages. Examples of such variables are: income, age, number of children, length of prison sentences, amount given to charity, number of sexual partners, how many drinks you had last month, how many visits you made to a physician last year, the concentration of mercury in water as measured in parts per million.

In all of the following cases we will be dealing with two variables.

· The independent variable, or cause, will be qualitative.

· The dependent variable will be quantitative.

We will analyze data for a simple quasi-experimental design – a simple before and after experiment with no control group. We will teach individuals techniques to help them memorize lists of words. To measure how well individuals can memorize words, a list of 20 words is read out to an individual. The individuals then have one minute to write downs as many of the words that they can remember. Individual will have scores ranging from 0 to 20. Question: What are the independent and dependent variables?

A random sample of 40 individuals is given a memorization pre-test. For quantitative data we average the scores of all the individuals. The average pre-test score is 7.3 words. The individuals are then given memorization training. After the training a post-test is carried out. The average post-test score is 14.3 words.

Diff_exp = Posttest_exp – Pretest_exp

Diff_exp = 14.3 – 7.3 = 7.0 words.

While there is certainly an increase in the number of words that people can memorize, we cannot be sure that these individuals wouldn’t have improved anyway. Perhaps they would have concentrated more after the pre-test. The pre-test alone, and not the training may be responsible for the improvement. To get around this problem, the researchers use a control group. The control group receives the pre-test, but not the training. The results are as follows:

	Pretest	Stimulus	Posttest
Experimental	7.3	Training	14.3
Control	6.8	None	9.9

The following calculations are made to determine if there is an experimental effect:

Diff_exp = Posttest_exp – Pretest_exp = 14.3 – 7.3 = 7.0

Diff_control = Posttest_control – Pretest_control = 9.9 – 6.8 = 3.1

Effect = Diff_exp – Diff_control = 7.0 – 3.1 = 3.9

Interpretation: On average the experimental group could remember 3.9 more words than the control group. We attribute this to the training. For those of you with a background in statistics, we would carry out a statistical test to determine if the differences between the two groups are statistically significant.

Example

Hypothesis: does hypnosis reduce the number of cigarettes that smokers smoke?

Measure: the number of cigarettes smoked in a day.

A sample of smokers is asked to record how many cigarettes they smoke in a day. The smokers then receive daily hypnosis sessions for a period of 12 days. A month later the smokers are asked to record how many cigarettes they smoke in a day. The following are the results:

Experimental group			Control group
Smoker ID	Number before hypnosis	Number after hypnosis	Smoker ID	Number before hypnosis	Number after hypnosis
1	18	18	11	14	18
2	9	5	12	21	21
3	25	0	13	16	16
4	12	12	14	9	5
5	16	6	15	32	24
6	4	0	16	12	12
7	17	14	17	18	14
8	13	0	18	11	11
9	5	4	19	6	6
10	13	13	20	6	0

Results:

· Average number of cigarettes smoked prior to hypnosis: 13.2

· Average number of cigarettes smoked after to hypnosis: 7.2

· Diff_exp = Posttest_exp – Pretest_exp =7.3 – 13.3 = -6.0

There was an average drop of 7 cigarettes smoked per day.

This may be due to the hypnosis.

It could be due to other factors that we haven’t controlled for. For example, the price of cigarettes may have gone up during the same period.

To check this we compare changes in the experimental group to changes in the control group.

· Average number of cigarettes smoked prior to hypnosis: 14.5

· Average number of cigarettes smoked after to hypnosis: 12.7

· Diff_control = Posttest_control – Pretest_control =12.7 – 14.5 = -1.8

The control groups also smoked less, on average 1.8 fewer cigarettes a day.

Effect = Diff_exp – Diff_control = -6.0 – -1.8 = -4.2

Hypnosis reduced cigarette smoking on average by 4.2 cigarettes a day.

Cases where there is no experimental effect

	Case 1		Case 2
	Pretest	Posttest	Pretest	Posttest
Experimental	13.2	13.2	13.2	7.2
Control	13.2	13.2	13.2	7.2

Case 1: Both groups start at the same level. There is no change in either group.

Case 2: Both groups start at the same level. Both groups end at the same level. The amount of change is identical for each group. There is no effect that can be attributed to the stimulus.

	Case 3		Case 4
	Pretest	Posttest	Pretest	Posttest
Experimental	13.2	7.2	13.2	7.2
Control	14.5	8.5	14.1	6.8

Case 3: The control group smokes on average more than the experimental group before the hypnosis sessions. Both groups drop in the average number of cigarettes smoked. The drop is on average 6.0 cigarettes for the experimental group, and 6.0 cigarettes for the control group. There is no effect that can be attributed to the hypnosis.

Case 4: The experimental group drops on average by 6.0, while the control group drops on average by 7.3 cigarettes. The control group dropped more than the experimental group. We cannot conclude that hypnosis reduced the number of cigarettes smoked.

Cases where there is an experimental effect

	Case 5		Case 6
	Pretest	Posttest	Pretest	Posttest
Experimental	13.2	7.2	13.2	7.2
Control	13.2	13.2	13.2	9.2

Case 5: Both the experimental and control groups smoked the same average number of cigarettes at the pre-test. The experimental group smoked fewer cigarettes after hypnosis, while the control group showed no change. Hypnosis does appear to reduce the number of cigarettes smoked.

Case 6 Both the experimental and control groups smoked the same average number of cigarettes at the pre-test. The experimental group smoked fewer cigarettes after hypnosis, while the control group also smoked fewer cigarettes. The drop was greater for the experimental group (6.0) than for the control group(4.0). Hypnosis does appear to reduce the number of cigarettes smoked.

	Case 7		Case 8
	Pretest	Posttest	Pretest	Posttest
Experimental	13.2	7.2	13.2	14.2
Control	13.2	13.7	14.1	17.8

Case 7 Both the experimental and control groups smoked the same average number of cigarettes at the pre-test. The experimental group smoked fewer cigarettes after hypnosis, while the control group smoked more cigarettes. Hypnosis does appear to reduce the number of cigarettes smoked.

Case 8 Both the experimental and control groups smoked more cigarettes at the pre-test. The increase was on average 1 cigarette a day more for the experimental group, and 3.7 cigarettes a day more for the control group. Hypnosis does appear to affect the number of cigarettes smoked. How can we state this given that they both went up? In this case hypnosis has reduced the increase, but not actually caused a drop in the number of cigarettes smoked. Such a situation might occur if the price of cigarettes dropped between the time of the pre-test and post-test.

Experimental group			Control group
Smoker ID	Number before hypnosis	Number after hypnosis	Smoker ID	Number before hypnosis	Number after hypnosis
1	18	18	11	14	18
2	9	5	12	21	21
3	25	0	13	16	16
4	12	12	14	9	5
5	16	6	15	32	24
6	4	0	16	12	12
7	17	14	17	18	14
8	13	0	18	11	11
9	5	4	19	6	6
10	13	13	20	6	0

	Case 1		Case 2
	Pretest	Posttest	Pretest	Posttest
Experimental	13.2	13.2	13.2	7.2
Control	13.2	13.2	13.2	7.2

	Case 3		Case 4
	Pretest	Posttest	Pretest	Posttest
Experimental	13.2	7.2	13.2	7.2
Control	14.5	8.7	14.1	6.8

	Case 5		Case 6
	Pretest	Posttest	Pretest	Posttest
Experimental	13.2	7.2	13.2	7.2
Control	13.2	13.2	13.2	9.2

	Case 7		Case 8
	Pretest	Posttest	Pretest	Posttest
Experimental	13.2	7.2	13.2	14.2
Control	13.2	13.7	14.1	17.8