Tag Archives: standard deviation

The Statistical t-Test


The statistical t-test is used to compare two conditions, specifically the means of two conditions. A t-test can be applied to both a between participants and within participants design. This test can only be done on normally distributed data, and as such is a parametric test. The purpose of the t-test is to decide whether or not the difference between the means of the two conditions is statistically significant. If the difference is statistically significant we are able to except our experimental hypothesis and also give some directionality to our hypothesis. If the difference between the means of the two conditions is not statistically significant we must reject our experimental hypothesis and accept the null hypothesis. The t-score is technically more than just the difference between the means. Just like normal data distribution, the t-score also has a 95% confidence interval, which means for the difference between the means to be statistically significant, the alpha level needs to be less than 0.05. The alpha level was decided on 0.05 to try and reduce the amount of type I and type II errors. Type I errors is when we reject the null hypothesis but should not have, and type II errors is when we reject the experimental hypothesis but we should not have. If the sample size is large and the null hypothesis is true, the distribution of the t-scores is also normal. The smaller the sample size becomes, the more tail-heavy the distribution becomes.

The way this is interpreted is if two groups come from the same population,  then 95% of the time, the t-score (reflecting the difference in the means) will be within the 95% area under the graph of the data.

Degrees of Freedom

Degrees of freedom for within-participants design is the same as the number of participants.

Degrees of freedom for between-participants design is the (number of participants in group 1 -1) + (number of Ps in group 2 -1)

SPSS will do the math for you!


When you start with your mean scores, assume that the null hypothesis is true and that there is no significant difference between the means.

Then set your significance level at p<0.05 or the alpha level, which is the same thing. SPSS should do this automatically.

Then using SPSS calculate the t-score.

If the t-score is within the 95% interval: accept the null hypothesis and reject the experimental hypothesis.

If the t-score is outside the 95% interval: reject the null hypothesis and accept the experimental hypothesis. You have now established that there is a significant difference between the two means.


This is an example of the type of output that will be given by SPSS. From this output you can answer the following questions:

Question 1: Is the experimental design within or between participants?

Answer: The experimental design is within. You can tell this from the heading where it says paired difference.

Question 2: What is the t-score?

Answer: The t-score is -9.60.

Question 3: What are the degrees of freedom?

Answer: The degrees of freedom (df) is 77.

Question 4: Is it two-tailed or one-tailed test?

Answer: It is two tailed as shown in the last box. Sig. (2-tailed).

Question 5: Is the result significant at an alpha level of 0.05? Why?

Answer: The result is significant at the alpha level because p<0.001, which obviously is less than 0.05.

 Reporting the Results 

This is an example of how you would report the following data for the results section of a lab report:

The mean and standard deviation of participants’ reaction time under conditions 1 and 2 are given in Table (not in this post). The data were analysed using a two-tailed within-participants t-test and an alpha level of 0.05.There is a statistically significant difference between the ideal IQ and the estimated IQ, with the estimated IQ significantly lower than IQ for an ideal job, t(77) = -9.60, p <0.001.

Dispersion and Central Tendency

I think people underestimate the amount of statistics that is necessary for psychological research. Luckily, as long as you understand the theory behind the statistics, most of the math is done by a computer. For my undergraduate course we use a programme called SPSS, which you can buy of amazon but most universities supply for a reduced price for their students. This post will be an introduction to the basics of statistical theory used in psychological research.



Levels of Measurement

Nominal measurement is the lowest level of measurement, and includes categorical data and measures of frequencies.

Ordinal measurement involves rating scales to measure participant responses.

Interval measurement involves equal intervals; for example, measuring temperature. Interval measurement has no absolute zero worth.

Ratio measurement involves intervals with an absolute zero worth.

N.B: parametric tests can only be used with interval or ratio measurement unless you convert the data into numerical values.

Types of Data Seen in Psychological Research

Continuous numerical data is data that can take any value within a certain range. The issue with continuous numerical data is that it is heavily dependent on the accuracy of measuring instrument.

For example: height, weight, reaction time

Discrete numerical data is data that can only take a specific value within a certain range. Questions that involve how many of something or the presence or absence of data is usually dealing with discrete numerical data.

For example: Numerical scores on a questionnaire: how many times have you been oversees?

Categorical data does not deal with a specific numerical value, but rather what group variables can be placed into. The issue with categorical data is that it can be too extreme. The people under one label can be very different from each other. Plus, it is very difficult to make appropriate intervals for categorical data.

For example: gender, nationality, etc.Categorical data can also come from continuos or discrete variables


Types of Statistics

Descriptive statistics summarise the properties of a sample of data usually through measures of central tendency and dispersion. Measures of central tendency include: mean, median and mode. Dispersion refers to the spread of data, providing information about the mean accuracy. Different measures of dispersion include: range, variance, and standard deviation.

Inferential statistics use the properties found from the descriptive statistics to make estimations of of the properties of the population.

Central Tendency 

The mean is the average score. The mode is the most frequently occurring score, and the median is the middle score (when points are organised from lowest to highest value). The mean is the preferred measure of central tendency because it takes into account all the data of the population. The problem with using the mean, however, is that is easily influenced by extreme scores.


Dispersion measured the spread of data and as mentioned above, gives the mean accuracy.

The most common approach for calculating the deviance is by calculating the variance. The variance is the: (sum of the squared deviances)/(the number of observations – 1). The unit is always the square of the measurement unit. The variances indicates how much the scores differ from one another.

N.B: The squared deviances is found by calculating the difference between each observation and the mean, and then squaring this value.

Standard deviation is the square root of the variance, and is much easier to deal with because it does not have squared units like the variance.



ZHENG, Y. (2013). Referencing and citation – Harvard style, from PSY104 Methods and Reasoning for Psychologists. University of Sheffield, Richard Roberts Building on 4th February. Available from: Blackboard.
[Accessed 4/02/13].