Critical values of the Chi-square (
2)
distribution for p = 0.05 & 0.01 The critical
value
for a single-classification
("either
/ or") 2 test (with degrees
of freedom = df = 1)
is 3.84 at p = 0.05.
That is, for a test of a
hypothesis with two alternative outcomes
(heads or tails, round or wrinkled), 2 = 3.84
or
greater indicates that the chances are 5% or less that so
large a
deviation could have been obtained simply by chance.
[Alternatively, in
a large numer of similar experiments, such a result would
occur about
one time in 20]. Given a 2 = 6.635, the chance is less
than 1%. A
statistical outcome that occurs
less than 5% of
the time is said to be significant and
designated *;
a results obtained less than 1% of the
time is highly
significant and designated **.
Non-significant outcomes are designated ns.
2 = 
[dev2/exp]
2 = 1/5 + 1/5 = 0.40, ns
2 test (with degrees
of freedom = df = 1)
is 3.84 at p = 0.05.
That is, for a test of a
hypothesis with two alternative outcomes
(heads or tails, round or wrinkled), 2 = 3.84
or
greater indicates that the chances are 5% or less that so
large a
deviation could have been obtained simply by chance.
[Alternatively, in
a large numer of similar experiments, such a result would
occur about
one time in 20]. Given a 2 = 6.635, the chance is less
than 1%. A
statistical outcome that occurs
less than 5% of
the time is said to be significant and
designated *;
a results obtained less than 1% of the
time is highly
significant and designated **.
Non-significant outcomes are designated ns.| obs |
exp |
dev |
dev2/exp |
|
| I |
a |
c |
a-c |
(a-c)2
/ c |
| II |
b |
d |
b-d |
(b-d)2 / d |
2 = [dev2/exp]Consider
two
experiments in which coins are tossed, and the two
possibilites are
heads (H) or tails (T). The first involves n = 10 tosses, the
second n = 100
tosses
| obs |
exp |
dev |
dev2/exp |
|
| H |
6 |
5 |
1 |
1 / 5 |
| T |
4 |
5 |
-1 |
1 / 5 |
2 = 1/5 + 1/5 = 0.40, ns| obs |
exp |
dev |
dev2/exp |
|
| H |
60 |
50 |
10 |
100 /
50 |
| T |
40 |
50 |
-10 |
100 /
50 |
2 = 2.0 + 2.0
=
4.0*, p <
0.05
[or 0.05 > p >
0.01]Note that the same 60%/40% proportion is not significantly different from random in the smaller experiment, but is significant in the larger. This emphasizes that actual numbers, not proportions, must be tested, and that statistical significance is more accurately estimated with larger experiments.