The results of these tests are only valid when the data are normally-distributed. If the data are not normally-distributed, use another statistical test, such as the Mann-Whitney U-test.

## Explanation of terms
The standard deviation ( |

**To test whether the mean of a sample () differs significantly from a predicted value ( µ)...**

Calculate the 'standard error of the mean' (SEM):

Calculate the t-statistic:

Use the table of critical values (below) to find out whether or not the result is significant.

**To test whether the mean of a sample ( _{1}) differs significantly from the mean of another sample (_{2})...**

Calculate the 'standard error of the mean' (SEM):

Calculate the t-statistic:

Use the table of critical values (below) to find out whether or not the result is significant.

Alternatively, type the following formula into *Microsoft Excel* (where `A1:A10` and `B1:B10` are the ranges of cells containing the data):

`=TTEST(A1:A10,B1:B10,2,3)`

This will give you a *p*-value (see below) indicating how significant the result is.

The table below gives the t-value at which the result has a particular level of 'significance'.

*d.f.* is the number of 'degrees of freedom'. In this case, *d.f.* = *n* -1

(If the exact *d.f.* value that you want is not included in the table, use the closest value below it that is included.)

*p* is the probability that the difference between two samples, or the difference between a sample and the theoretical result, is entirely due to chance.

d.f. | p=0.1 | p=0.05 | p=0.01 |
---|---|---|---|

2 | 2.92 | 4.30 | 9.92 |

3 | 2.35 | 3.18 | 5.84 |

4 | 2.13 | 2.78 | 4.60 |

5 | 2.02 | 2.57 | 4.03 |

6 | 1.94 | 2.45 | 3.71 |

7 | 1.89 | 2.36 | 3.50 |

8 | 1.86 | 2.31 | 3.36 |

9 | 1.83 | 2.26 | 3.25 |

10 | 1.81 | 2.23 | 3.17 |

11 | 1.80 | 2.20 | 3.11 |

12 | 1.78 | 2.18 | 3.05 |

13 | 1.77 | 2.16 | 3.01 |

14 | 1.76 | 2.14 | 2.98 |

15 | 1.75 | 2.13 | 2.95 |

16 | 1.75 | 2.12 | 2.92 |

17 | 1.74 | 2.11 | 2.90 |

18 | 1.73 | 2.10 | 2.88 |

19 | 1.73 | 2.09 | 2.86 |

20 | 1.72 | 2.09 | 2.85 |

21 | 1.72 | 2.08 | 2.83 |

22 | 1.72 | 2.07 | 2.82 |

23 | 1.71 | 2.07 | 2.81 |

24 | 1.71 | 2.06 | 2.80 |

25 | 1.71 | 2.06 | 2.79 |

26 | 1.71 | 2.06 | 2.78 |

27 | 1.70 | 2.05 | 2.77 |

28 | 1.70 | 2.05 | 2.76 |

29 | 1.70 | 2.05 | 2.76 |

30 | 1.70 | 2.04 | 2.75 |

35 | 1.69 | 2.03 | 2.72 |

40 | 1.68 | 2.02 | 2.70 |

45 | 1.68 | 2.01 | 2.69 |

50 | 1.68 | 2.01 | 2.68 |

60 | 1.67 | 2.00 | 2.66 |

70 | 1.67 | 1.99 | 2.65 |

80 | 1.66 | 1.99 | 2.64 |

90 | 1.66 | 1.99 | 2.63 |

100 | 1.66 | 1.98 | 2.63 |

Infinity | 1.64 | 1.96 | 2.58 |

**Example:** suppose that a t-test on a sample of 10 individuals (*d.f.* = 9) produced a *t*-value of 3.0. The table tells us that *p* is between 0.01 and 0.05 in this case (*p*=0.05 when *t*=2.26, and *p*=0.01 when *t*=3.25; our t-value lies in between these two). Therefore, the probability of the result arising by chance is less than 5% (*p*<0.05), so this is a fairly significant result.