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Students T Test Sample Definition Essays

'Student's' t Test (For Independent Samples)

Use this test to compare two small sets of quantitative data when samples are collected independently of one another. When one randomly takes replicate measurements from a population he/she is collecting an independent sample. Use of a paired t test, to which some statistics programs unfortunately default, requires nonrandom sampling (see below).


  • Only if there is a direct relationship between each specific data point in the first set and one and only one specific data point in the second set, such as measurements on the same subject 'before and after,' then the paired t test MAY be appropriate.
  • If samples are collected from two different populations or from randomly selected individuals from the same population at different times, use the test for independent samples (unpaired).
  • Here's a simple check to determine if the paired t test can apply - if one sample can have a different number of data points from the other, then the paired t test cannot apply.


'Student's' t Test is one of the most commonly used techniques for testing a hypothesis on the basis of a difference between sample means. Explained in layman's terms, the t test determines a probability that two populations are the same with respect to the variable tested.

For example, suppose you collected data on the heights of male basketball and football players, and compared the sample means using the t test. A probability of 0.4 would mean that there is a 40% liklihood that you cannot distinguish a group of basketball players from a group of football players by height alone. That's about as far as the t test or any statistical test, for that matter, can take you. If you calculate a probability of 0.05 or less, then you canreject the null hypothesis (that is, you can conclude that the two groups of athletes can be distinguished by height.

To the extent that there is a small probability that you are wrong, you haven't proven a difference, though. There are differences among popular, mathematical, philosophical, legal, and scientific definitions of proof. I will argue that there is no such thing as scientific proof. Please see my essay on that subject. Don't make the error of reporting your results as proof (or disproof) of a hypothesis. No experiment is perfect, and proof in the strictest sense requires perfection.

Make sure you understand the concepts of experimental error and single variable statistics before you go through this part. Leaves were collected from wax-leaf ligustrum grown in shade and in full sun. The thickness in micrometers of the palisade layer was recorded for each type of leaf. Thicknesses of 7 sun leaves were reported as: 150, 100, 210, 300, 200, 210, and 300, respectively. Thicknesses of 7 shade leaves were reported as 120, 125, 160, 130, 200, 170, and 200, respectively. The mean ± standard deviation for sun leaves was 210 ± 73 micrometers and for shade leaves it was158 ± 34 micrometers. Note that since all data were rounded to the nearest micrometer, it is inappropriate to include decimal places in either the mean or standard deviation.

For the t test for independent samples you do not have to have the same number of data points in each group. We have to assume that the population follows a normal distribution (small samples have more scatter and follow what is called a t distribution). Corrections can be made for groups that do not show a normal distribution (skewed samples, for example - note that the word 'skew' has a specific statistical meaning, so don't use it as a synonym for 'messed up').

The t test can be performed knowing just the means, standard deviation, and number of data points. Note that the raw data must be used for the t test or any statistical test, for that matter. If you record only means in your notebook, you lose a great deal of information and usually render your work invalid. The two sample t test yields a statistic t, in which

X-bar, of course, is the sample mean, and s is the sample standard deviation. Note that the numerator of the formula is the difference between means. The denominator is a measurement of experimental error in the two groups combined. The wider the difference between means, the more confident you are in the data. The more experimental error you have, the less confident you are in the data. Thus the higher the value of t, the greater the confidence that there is a difference.

To understand how a precise probability value can be attached to that confidence you need to study the mathematics behind the t distribution in a formal statistics course. The value t is just an intermediate statistic. Probability tables have been prepared based on the t distribution originally worked out by W.S. Gossett (see below). To use the table provided, find the critical value that corrresponds to the number of degrees of freedom you have (degrees of freedom = number of data points in the two groups combined, minus 2). If t exceeds the tabled value, the means are significantly different at the probability level that is listed. When using tables report the lowest probability value for which t exceeds the critical value. Report as 'p < (probability value).'

In the example, the difference between means is 52, A = 14/49, and B = 3242.5. Then t = 1.71 (rounding up). There are (7 + 7 -2) = 12 degrees of freedom, so the critical value for p = 0.05 is 2.18. 1.71 is less than 2.18, so we cannot reject the null hypothesis that the two populations have the same palisade layer thickness. So now what? If the question is very important to you, you might collect more data. With a well designed experiment, sufficient data can overcome the uncertainty contributed by experimental error, and yield a significant difference between samples, if one exists.

If you have lots of data and the probability value becomes smaller but still does not reach the 'magic' number 0.05, should you keep collecting data until it does? At this point, consider the biological significance of the question. If you did find adifference of 0.1% between palisade layers of sun and shade leaves respectively, just how important could it be?

When reporting results of a statistical analysis, always identify what data sets you compared, what test was used, and for most quantitative data report mean, standard deviation, and the probability values. Make sure the outcome of the analysis is clearly reported. Some spreadsheet programs include the t test for independent variables as a built-in option. Even without a built-in option, is is so easy to set up a spreadsheet to do a paired t test that it may not be worth the expense and effort to buy and learn a dedicated statistics software program, unless more complicated statistics are needed.

Historical note

You may be wondering where the name 'Student' came from, and why the quotation marks. The basis of the t test would be known as 'Gosset's t distribution' if it werent for contractural obligations that prevented W.S. Gosset from taking credit for its development. Gosset used measurements of the heights and left middle finger lengths of criminals in a local prison to work out the t distribution empirically. The mathematical theory followed. Gosset published his distribution in 1908 under the pseudonym 'Student.'

Statwing represents t-test results as distribution curves. Assuming there is a large enough sample size, the difference between these samples probably represents a “real” difference between the populations from which they were sampled.

Note: The below discusses the unranked “independent samples t-test”, the most common form of t-test.


A t-test helps you compare whether two groups have different average values (for example, whether men and women have different average heights).


Let’s say you’re curious about whether New Yorkers and Kansans spend a different amount of money per month on movies. It’s impractical to ask every New Yorker and Kansan about their movie spending, so instead you ask a sample of each—maybe 300 New Yorkers and 300 Kansans—and the averages are $14 and $18. The t-test asks whether that difference is probably representative of a real difference between Kansans and New Yorkers generally or whether that is most likely a meaningless statistical fluke.

Technically, it asks the following: If there were in fact no difference between Kansans and New Yorkers generally, what are the chances that randomly selected groups from those populations would be as different as these randomly selected groups are? For example, if Kansans and New Yorkers as a whole actually spent the same amount of money on average, it’s very unlikely that 300 randomly selected Kansans each spend exactly $14 and 300 randomly selected New Yorkers each spend exactly $18. So if you’re sampling yielded those results, you would conclude that the difference in the sample groups is most likely representative of a meaningful difference between the populations as a whole.


A t-test asks whether a difference between two groups’ averages is unlikely to have occurred because of random chance in sample selection. A difference is more likely to be meaningful and “real” if
(1) the difference between the averages is large,
(2) the sample size is large, and
(3) responses are consistently close to the average values and not widely spread out (the standard deviation is low).

The t-test’s statistical significance and the t-test’s effect size are the two primary outputs of the t-test. Statistical significance indicates whether the difference between sample averages is likely to represent an actual difference between populations (as in the example above), and the effect size indicates whether that difference is large enough to be practically meaningful.

The “One Sample T-Test” is similar to the “Independent Samples T-Test” except it is used to compare one group’s average value to a single number (for example, do Kansans on average spend more than $13 per month on movies?). For practical purposes you can look at the confidence interval around the average value to gain this same information.

The “paired t-test” is used when each observation in one group is paired with a related observation in the other group. For example, do Kansans spend more money on movies in January or in February, where each respondent is asked about their January and their February spending? In effect a paired t-test subtracts each respondent’s January spending from their February spending (yielding the increase in spending), then take the average of all those increases in spending and looks to see whether that average is statistically significantly greater than zero (using a one sample t-test).

The “ranked independent samples t-test” asks a similar question to the typical unranked test but it is more robust to outliers (a few bad outliers can make the results of an unranked t-test invalid).

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