This series of web pages covers the topics of Research Methods and Statistical Analyses in Psychology. These topics build upon the Research Methods section covered in undergraduate General Psychology courses. It is helpful to review those topics covered in my AP Psychology Self-Study Hub.
📘Overview of Data Analyses and Descriptive Statistics
📘Hypothesis Testing, Probability, Power, Normal Distribution and Z test
📘T-Tests for Single Sample, Paired-Sample and Independent Samples
📘One-Way Analyses of Variance Factorial Analysis of Variance
Overview of Data Analyses and Descriptive Statistics
Let’s start with an example of experimental research. Research starts with an issue or question.
Example 1: To Invest or not to Invest
My question is what is the impact of buying stocks? Does it make people richer or poorer? As a researcher, I have a theory to test: Investing money in the stock market affects people’s financial situation. I am not saying if it makes investors richer or poorer.
From the theory the following Hypothesis is derived: Between two groups of people starting with the same amount of money in their possession, the group that have been randomly assigned to invest in the stock market should have different amount of money left than the group that has been randomly assigned not to invest.
The researcher conducted the following Experiment to test the hypothesis:
He recruited 8 volunteers to participate in the study, which lasted for six months. All participants received the same amount of funds at the beginning of the study.
He randomly assigned 4 of the participants to a group that must invest their funds in the stock market during the six-month period. He randomly assigned the other 4 participants to a group that did not invest in the same period. The researcher measured how much money each participant still had in his or her possession at the end of the period.
Do you intend to show that the hypothesis to be true only for the people you have observed, or for all people who invest or not invest in the stock market?
A population is the total set of possible cases of observation (e.g. people, objects, events and etc.) with a particular characteristic.
In this example, I am interested in two populations: all people who invest in the stock market, and all people who do not invest.
If my theory is true, there should be a significant difference between these two populations in their financial situations. This is a prediction based on my theory. It is called a research hypothesis.
What should I do to verify this hypothesis with certainty? I could measure all members of the two populations. That would be a “census.”
Is that possible in this case with investors and non-investors? No. What should you do then?
A sample is a restricted number of cases of actual observations. In the example, I have two samples: A sample of 4 investors and a sample of 4 non-investors. These are my cases of actual observations, people whom I have observed.
Statistical Methods: Descriptive Versus Inferential Statistics
The field of statistics has two branches, descriptive and inferential statistics.
Descriptive statistics are tools, such as tables, graphs and calculated values for organizing and summarizing information about a collection of actual observations. In other words, the values I compute from the subjects I have actually observed are descriptive statistics.
Descriptive statistics tell you about the values of the samples you have observed on the variables.
Inferential statistics are tools that enable the researcher to infer whether they are justified to generalize the conclusion beyond to the samples, to the entire populations of interest. Can we make the same conclusion about the financial situation of all investors and non-investors? In real life it is often impossible to observe the entire population. Any statement we make about the population is an inference, rather than description of observed facts.
Descriptive Statistics
Descriptive statistics include two broad types, the Central Tendency and Dispersion. Central tendency includes the mean, median, and mode. Dispersion includes the range, variance, and standard deviation.
1. Center or central tendency, including the mean, median, and mode
An experiment yields raw data, or the scores of the subjects on the dependent variable. Descriptive statistics offer tools for summarizing research data. The most commonly used group information in research is the mean, or the average of the group of numbers we have. The mean is an index of central tendency. The central tendency is the most typical or representative value of a group of scores.
The mean is calculated by adding up all the scores in the group, and then dividing the sum by the total number of scores in that group.
Here is the mathematical formula:
M = ΣX/N
Another way of representing the central tendency of a set of scores is the median. Median: it is the middle score of the group. The following are the steps for getting the median.
(1) Line up all the scores in the group from the lowest to the highest. (2) Find the score right in the middle. This is the median. If you have an even number of scores in your sample, you will have two scores right there in the middle. In this case the average of the two middle scores is the median.
Another way of representing the central tendency of a set of scores is the mode. The mode is the most frequent score in the group. To get the mode you must count how many times a particular score occurs. The score that occurs the most frequently is the mode.
Let’s go back to the example.
Investors
| Participant # | Funds left (K dollars) | X-M | (X-M)2 |
| 1 | 5 | 5 – 6 = -1 | 1 |
| 2 | 10 | 10-6 = 4 | 16 |
| 3 | 8 | 8 – 6 = 2 | 4 |
| 4 | 1 | 1 – 6 = -5 | 25 |
| Sum | 24 | ||
| MInvestor | 24/4 = 6 | ||
| Sum of Squared Deviation (SSInvestor) | 1 + 16 + 4 + 25 = 46 | ||
VarianceInvest = SSInvest /(N-1) = 46/3 = 15.33
N-1 = degrees of freedom
Standard deviation = square root of Variance = √15.33 = 3.91
Non-Investors
| Participant # | Funds left (K dollars) | X-M | (X-M)2 |
| 5 | 6 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 7 | 6 | 0 | 0 |
| 8 | 6 | 0 | 0 |
| Sum | 24 | 0 | 0 |
| MNon-invest | 24/4 = 6 | 0 | 0 |
| Sum of Squared Deviation (SSInvest) | |||
SS = 0; Variance = SS/(N-1) = 0/3 = 0; standard dev. = 0
2. Spread or dispersion: variance and standard deviation
Besides knowing the most typical or representative value among the scores, it is also useful to know how spread out the scores are in the group, or how the score differ from each other. A measure of how spread out the scores is called dispersion, a type of descriptive statistics.
One measure of spread is the range, the difference between the maximum and minimum.
A more detailed measure of spread is variance and standard deviation.
Why do we need to know dispersion?
If you only look at the means of the investors and non-investors, you may think that buying stocks has no effect on people’s financial situation. But if you look at the dispersion of the two groups, you come to a different conclusion.
Example two: Social learning and aggression
Albert Bandura did an experiment on how violence on TV programs influence children’s aggressive behavior. His theory was that kids are more likely to imitate aggressive behavior they see on TV if the modeled aggressive behavior is rewarded, rather than ignored. He predicts a relationship between the variable of reward for modeled aggressive behavior and the variable of kids’ tendency to imitate this behavior. Here is an imaginary scenario of the study.
We have a total of 6 kids. We assign 3 of them to treatment condition 1, and the other 3 to treatment condition 2. In condition 1, the 3 kids watched a video in which a person performed aggressive behavior (hitting a life-size doll with a toy hammer), and then got praised by another person for this aggressive behavior. In condition two, the 3 kids watched a video in which the same person performed the same aggressive behavior but did not get praised by another person. Then, each kid in both conditions was let to stay in a room with the life-size doll he/she had seen on TV for a period of time, and given a toy hammer to play with. The experimenter recorded the number of times each kid hit the doll with the hammer during this period of time. The following are data collected from the 6 participants.
| subject # | group | hits |
| 1 | 1 | 4 |
| 2 | 1 | 6 |
| 3 | 1 | 5 |
| 4 | 2 | 0 |
| 5 | 2 | 2 |
| 6 | 2 | 1 |
Group 1 consists of those who have seen the video in which the aggressive behavior was rewarded. Group 2 consist of those who have seen the video in which the aggressive behavior was not rewarded.
1. Calculate the mean, variance and standard deviation for each of the two individual groups.
| Sub # | X | M | X – M | (X – M)2 | Z = (X – M)/SD |
| 1 | 4 | 5 | -1 | 1 | (4-5)/1 =-1 |
| 2 | 6 | 5 | 1 | 1 | (6-5)/1 = 1 |
| 3 | 5 | 5 | 0 | 0 | (5-5)/1 = 0 |
sum of squared deviation (SS) = 2 Mean (M) = ΣX/N = (4 + 5 + 6)/3 = 5
Variance = SS/（N-1） = 2/2 = 1 Standard Dev. (SD) = Variance = √1 = 1
Group 2 (those who saw un-rewarded aggression)
| Sub# | X | M | X – M | (X – M)2 | Z |
| 4 | 0 | 1 | -1 | 1 | (0 – 1)/1 = -1 |
| 5 | 2 | 1 | 1 | 1 | (2 – 1)/1 = 1 |
| 6 | 1 | 1 | 0 | 0 | (1-1)/1=0 |
Mean (M) = ΣX/N = (0 + 2 + 1)/3 = 1
sum of squared deviation (SS) = 2 Variance (SD2)= SS/（N-1） = 2/1 = 1 Standard Dev. (SD) = √Variance = √1 = 1
Z-Scores
The central tendency statistics (mean, median and mode) and dispersion statistics (variance and standard deviation) give information about the group, the group of subjects in the experiment. In calculating the mean, the variance and the standard deviation, we abstract away from the individual scores. With Z scores, we are going back to individual subjects, to find out where each subject stands within the entire group on that particular variable.
The Z score is easy to calculate. Just take each raw score, subtract from it the mean and divide the difference by the standard deviation.
The formula: Z = (X – M)/SD.
As an example, we can calculate the Z score for each subject when we take the total set of subjects involved in the experiment as a single group. The following are made-up data on the Bandura modeling and aggression study, with the two groups combined.
| Sub# | X | M | X – M | Z score |
| 1 | 4 | 3 | 1 | |
| 2 | 6 | 3 | 3 | |
| 3 | 5 | 3 | 2 | |
| 4 | 0 | 3 | – 3 | |
| 5 | 2 | 3 | -1 | |
| 6 | 1 | 3 | -2 |
When we convert a group of raw scores into z scores, the resultant z scores always have a group mean of 0 and a standard deviation of 1.
Why do we need a z score for each subject? The z scores are called “standardized scores”. They have the same unit of increment. They represent the relative standing on a variable.
The Median and Inter Quartile Range (IQR)
When a data set has outliers or extreme values, the mean is not a good representation of the central tendency. For example, you ask a group of nine individuals to report how many pairs of shoes they each own and get the following data: 1 pair, 2 pairs, 3 pairs, 4 pairs, 5 pairs, 5 pairs, 5 pairs, 6 pairs, 100 pairs. The last person is an outlier.
Mean=(1+2+3+4+5+5+5+6+100)/9=131/9=14.56
The mean is approximately 14.56 pairs. However, eight out of nine individuals own 6 pairs or fewer. Therefore, 14.56 does not represent the typical individual very well.
The median is a better representation of the typical individual than the mean. There are nine observations, the median is the fifth observation, median = 5. The mean is sensitive to extreme scores (outliers). The median is resistant to extreme scores. When a distribution is highly skewed or contains extreme outliers, the median often provides a better representation of the typical observation.
When a data set contains outliers, variability is often best summarized by the interquartile range (IQR), which measures the spread of the middle 50% of the data. The IQR is the difference between the first and third quartiles:
First Quartile (Q1): The value below which 25% of the data falls.
Third Quartile (Q3): The value below which 75% of the data falls (or above which 25% of the data falls).
Quartiles are determined using the same method as the median, but applied separately to the lower and upper halves of the ordered data set. The interquartile range is calculated as:
IQR = Q3 – Q1
Example 3: Quartiles and IQR with Even Sample Size:
Consider a sample of 10 individuals (n = 10), the age at which each of them passed away is listed below:
62, 63, 64, 64, 70, 72, 76, 77, 81, 81
The median age of demise is 71, meaning 50% of the values are above 71, and 50% are below. To determine the quartiles, we divide the ordered data into a lower half and an upper half. Q1 is calculated as the median of the lower half, and Q3 is the median of the upper half. The lower half contains five values: 62, 63, 64, 64, 70. The median of these five is 64, which is Q1 or first quartile. The upper half contains five values: 72, 76, 77, 81, 81, the median of which is 77, or Q3 or third quartile.
The interquartile range is 77 – 64 = 13; the interquartile range is the range of the middle 50% of the data.
Example 4: Quartiles and IQR with an Odd Number Sample Size:
When the sample size is an odd number, the median and quartiles are determined in the same way. Suppose in the previous example, the lowest value (62) were excluded, and the sample size was n=9, the median is the middle number 72.
63, 64, 64, 70, 72, 76, 77, 81, 81
The quartiles are determined by looking at the lower and upper halves. There are 4 values in the lower half, the first quartile is the mean of the 2 middle values in the lower half ((64+64)/2=64). The same approach is used in the upper half to determine the third quartile ((77+81)/2=79).
Outliers and Tukey Fences:
When there are no outliers in a sample, the mean and standard deviation are used to summarize a typical value and the variability in the sample, respectively. When there are outliers in a sample, the median and interquartile range are used to summarize a typical value and the variability in the sample, respectively.
Tukey Fences: There are several methods for determining outliers in a sample. A very popular method is based on the following:
Outliers are values below Q1-1.5(Q3-Q1) or above Q3+1.5(Q3-Q1) or equivalently, values below Q1-1.5 IQR or above Q3+1.5 IQR. These are referred to as Tukey fences. For the diastolic blood pressures, the lower limit is 64 – 1.5 (77-64) = 44.5 and the upper limit is 77 + 1.5(77-64) = 96.5.
The diastolic blood pressures range from 62 to 81. There are no outliers. The best summary of a typical diastolic blood pressure is the mean (in this case 71.3) and the best summary of variability is given by the standard deviation (s=7.2).
Exercise:
Calculate the mean, variance and standard deviation for each of the two individual groups.
| Sub # | X | M | X – M | (X – M)2 | Z = (X – M)/SD |
| 1 | 4 | ||||
| 2 | 6 | ||||
| 3 | 5 |
Mean (M) =
sum of squared deviation (SS) =
Variance = SS/（N-1） =
Standard Dev. (SD) = Variance =
Group 2 (those who saw un-rewarded aggression)
| Sub# | X | M | X – M | (X – M)2 | Z |
| 4 | 0 | ||||
| 5 | 2 | ||||
| 6 | 1 |
Mean (M) =
sum of squared deviation (SS) = Variance (SD2)=
Standard Dev. (SD) =