t= ×N(^1) ⁄ 2
(3)
x
dbeing the mean difference between paired values and sdthe estimated
standard deviation of the differences.
Comparison of one experimental mean with a known value, m
t= ×N
(^1) ⁄ 2
(4)
Using the appropriate equation, an experimental value, texptl, is calculated and
compared with a tabulated value, ttab, at a defined probability level, usually
between 90 and 99%, and for N- 1 degrees of freedom (equations (3) and (4)) or
(N +M - 2 ) degrees of freedom (equation (1)). If texptlis less than ttab, then the null
hypothesis that there is no significant difference between the two experimental
means or between the experimental mean and a known value is accepted, i.e.
there is no evidence of a bias. However, if texptlis greater than ttab, there is a
significant difference indicating a bias.
Both one-tailed and two-tailed t-tests can be used, depending on circum-
stances, but two-tailed are often preferred (Table 3). The application of all three
t-test equations is demonstrated by the following examples.
Table 3. Critical values of t at the 95% and 99% (P=0.05 and 0.01) levels for a
two-tailed test
Number of degrees of freedom 95 percent level 99 percent level
2 4.30 9.92
5 2.57 4.03
10 2.23 3.10
18 2.10 2.88
Example 1
Two methods for the determination of polyaromatic hydrocarbons in soils were
compared by analyzing a standard with the following results:
No. of determinations by each method: 10
No. of degrees of freedom: 18
UV spectrophotometry: x
=28.00 mg kg-^1 s=0.30 mg kg-^1
Fluorimetry: x
=26.25 mg kg-^1 s=0.23 mg kg-^1
Do the mean results for the two methods differ significantly?
Equation (2) is first used to calculate a pooled standard deviation:
spooled= (N−1)s^2 A+(M−1)s^2 B / N+M− (^2)
(^1) ⁄ 2
={(9 ¥0.3^2 + 9 ¥0.23^2 )/18}
(^1) ⁄ 2
spooled=0.267 mg kg−^1
Then equation (1) is used to evaluate texptl
texptl= ×
(^1) ⁄ 2
={(28.0 -26.25)/0.267} ¥ 5
(^1) ⁄ 2
=14.7
For 18 degrees of freedom, the two-tailed value of ttabat the 95% probability
level is 2.10, and at the 99% level it is 2.88.
NM
N+M
(x
A−x
B)
spooled
(x
−m)
s
x
_
d
sd
38 Section B – Assessment of data