Statistics Tutorial

P-Values and T-Tables

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Getting Started	General Instructions \| Introduction to Your Study \| Experimental Design \| Stating a Hypothesis
Descriptive Statistics	Histograms \| Central Tendency \| Standard Deviation \| Confidence Intervals
Comparing Two Samples	Samples and Populations \| Choosing a T-Test \| Paired T-Test \| P-Values and T-Tables
Important Concepts	The Normal Distribution \| Z Scores \| Probability Distributions
Levels	You are currently on P-Values and T-Tables at level 2. Level 1 \| Level 2 \| Level 3
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Explanation

Using T-Tables
We have learned the following things about a t-test:

The t-test produces a single value, t, which grows larger as the difference between the means of two samples grows larger;
t does not cover a fixed range such as 0 to 1 like probabilities do;
You can convert a t-value into a probability, called a p-value;
The p-value is always between 0 and 1 and it tells you the probability of the difference in your data being due to sampling error;
The p-value should be lower than a chosen significance level (0.05 for example) before you can reject your null hypothesis.

Converting from t-values to p-values is usually done by software but you can do it by hand by looking values up in a table. This page explains how.

T-tables are filled with t-values. Each t-value is in a column that is specific to a given significance level. The t-values shown in the table are known as critical values. If your t-value is greater than or equal to the critical value in the table, then you can conclude that your p-value is less than the significance level you have chosen. The procedure is simple. Once you have chosen your significance level you look to see whether your t-value is larger than the critical t-value shown in the column relating to your chosen significance level. If it is, then you can say that p is less than your chosen significance level.

The t-table has more than one row of t-values. Each row corresponds to a given number of degrees of freedom. Degrees of freedom are explained at level on of this topic. Use the row of the table that corresponds to the degrees of freedom in your data.

The final thing that you need to know about using t-tables is that you must read them differently depending on whether your test is one-tailed or two-tailed. The rule is simple enough:

The p-value for a two tailed test is twice what it would be for a one tailed test.

Some tables (such as the ones on this page) show a p-value heading for both one and two-tailed tests to make it easier. If your test is one tailed, find the p-value you need in the top row. If your test is two tailed, find your p-value in the second row. Notice how the p-values for the two tailed test are simply double those for the one tailed test.

Unfortunately, some tables show only one-tailed values and some show only two-tailed values. If you are using a table from a book or from the internet, make sure you know what it shows. You can easily convert from one to two or two to one by remembering the p for two-tailed tests is twice what it is for one-tailed tests.

Reporting p-Values and t-Values
You report the results of a t-test in the following way:

t(df)=t, p<p

Where df is the degrees of freedom of your data, t is the t-value you found and p is the p-value you found.

Exploration

Converting from a t-value to a p-value requires some tricky maths, so statisticians use pre-calculated tables to make it easy. These tables are called t-tables.

A t-table is shown below. Notice that it has a number of columns, each showing a different significance level (p). T-tables do not tell you the exact value of p. They list a few key values of p and tell you what value of t is required to produce a p-value less than the listed value. It has many rows and each row is marked with a number showing the degrees of freedom (df). Once you have chosen your significance level, p, calculated a value for t, and worked out how many degrees of freedom you have, you can find the entry in the t-table that you need as follows:

Look down the column that corresponds to your chosen value for p
Find the row that corresponds to your degrees of freedom, and where they meet, you will find the value you need.

This value is called the critical value. The final thing to do is compare this value with your value of t

If your t-value is greater than or equal to this value, then t is significant and you have found a difference
If your t-value is less than this value is then t is not significant.

Here are some examples for you to work out using the table below.

p<0.05

df=4

t=3.143
Tails=1
Critical value:

Significance:

p<0.01

df=12

t=3.143
Tails=1
Critical value:

Significance:

p<0.05

df=8

t=3.143
Tails=2
Critical value:

Significance:

p<0.01

df=7

t=3.143
Tails=2
Critical value:

Significance:

One-tailed p	0.1	0.05	0.025	0.01	0.005
Two-tailed p df	0.2	0.1	0.05	0.02	0.01
1	3.078	6.314	12.706	31.821	63.657
2	1.886	2.92	4.303	6.965	9.925
3	1.638	2.353	3.182	4.541	5.841
4	1.533	2.132	2.776	3.747	4.604
5	1.476	2.015	2.571	3.365	4.032
6	1.44	1.943	2.447	3.143	3.707
7	1.415	1.895	2.365	2.998	3.499
8	1.397	1.86	2.306	2.896	3.355
9	1.383	1.833	2.262	2.821	3.25
10	1.372	1.812	2.228	2.764	3.169
11	1.363	1.796	2.201	2.718	3.106
12	1.356	1.782	2.179	2.681	3.055
13	1.35	1.771	2.16	2.65	3.012
14	1.345	1.761	2.145	2.624	2.977
15	1.341	1.753	2.131	2.602	2.947

Application

Now lets look at your own data. The first step is to decide on the number of degrees of freedom you have. This depends on whether your samples are paired or independent.

Your experiment compares when is Pre-Test with when is Post-Test. is measured from the same People under both conditions, Pre-Test and Post-Test.
You measured 30 People in each condition.
Your test is one-tailed.

What is your experimental design?

Should you be looking in the one-tailed or two-tailed column?

How many degrees of freedom does your data have?

Your t-value is -2.13. We will ignore the minus sign and just use 2.13, as the values in the t-tables are all positive. Using a p-value of 0.05, look in the section of the t-table shown below and find the critical value for your data.

One-tailed p 0.1 0.05 0.025 0.01 0.005
Two-tailed p
df 0.2 0.1 0.05 0.02 0.01
24 1.318 1.711 2.064 2.492 2.797
25 1.316 1.708 2.06 2.485 2.787
26 1.315 1.706 2.056 2.479 2.779
27 1.314 1.703 2.052 2.473 2.771
28 1.313 1.701 2.048 2.467 2.763
29 1.311 1.699 2.045 2.462 2.756
30 1.31 1.697 2.042 2.457 2.75
31 1.309 1.696 2.04 2.453 2.744
32 1.309 1.694 2.037 2.449 2.738
33 1.308 1.692 2.035 2.445 2.733
34 1.307 1.691 2.032 2.441 2.728
p<0.05
df=29
t=2.13
Tails=1
Critical value:
Significance:
Which of these would be the correct way to report this result?
Now look up your t-values at the 0.01 level.
p<0.01
df=29
t=2.13
Tails=1
Critical value:
Significance:
Which of these would be the correct way to report this result?

Paired T-Test | The Normal Distribution