Sign Test

Sign and Binomial Test

The Sign and Binomial tests assess precisely the likelihood of an either/or result; the probability of getting a particular result if the two alternatives are considered, say, as faces of a coin. The difference between the two tests is that:

the Binomial test is best considered as a two-category, one-way Chi-square test: literally a comparison of the number of individuals 'Agreeing' or 'Disagreeing' with an attutude statement with the null hypothesis which assumes an equal number in each category.

The Sign test is identical in how it assesses probability. It differs in that it studies the difference between, say, response at the beginning and end of a course. A practical example of this is illustrated in

Tall, G. & Grove, J. (1993) A Study of the Effectiveness of the Religious Education in Early Years Project Inservice Course. Panorama: International Journal of Comparative Religious Education and Values, 1, 5, 101-115 ISSN 0937 8219

In this article the researchers assessed the views of teachers taking a religious education course both when they started the course and when they finished it. The mean response of the students on each attitude statement was calculated and if, at the end of the course, the change was in line with the views of the people running it then the difference was noted as +ve; if the difference was not in line then the difference was noted as -ve. The total number of differences (T = +ve plus -ve) can be considered as being equivalent to the number of times a coin is tossed. The smaller number of the +ve or -ve differences can be considered as being equivalent to the number of, say, heads. Significance in the sign test is simply the probability of getting just that number of heads when a coin is tossed T times.

I Assessing Probability in Sign and Binomial Tests - Pascal’s Triangle

Whenever a coin is tossed on to a flat surface it either lands with the face (H: heads) or the pattern (T: tails) uppermost.

The probability of a number of coins landing in a particular combination can be quickly calculated using figure 1, which is known as Pascal's triangle. If a coin lands purely by chance, the following possible patterns of heads and tails will occur:

Figure 1 Pascal's Triangle:

Since the coin must land either heads or tails, then:

In toss 1 there are only two possibilities 1Head or 1Tail.

On toss 2 the coin that landed Heads can either land Heads again (1HH) or Tails (1HT); similarly the coin that landed Tails must land either Heads (1HT) or Tails (1TT) Making a total of 1HH, 2HT, 1TT. Thus there are 4 possible sequences (HH, HT, TH, TT) but only 3 alternative categories (HH, One head and one tail, TT):

Note: in each category the sequences differ, thus in row 6 the 6 tosses with 5 heads and 1 tail are:

HHHHHT HHHHTH HHHTHH HHTHHH HTHHHH & THHHHH

What Does Pascal's Triangle Tell Us About the Probability of Getting All Heads OR All Heads & All Tails:

a) when a coin is tossed 5 times:

the probability of getting all heads is 1 in 32 3.125%

the probability of getting all heads or all tails is 2 in 32 6.25%

There is a 6.25% chance of obtaining all heads or all tails. Since the level of probability required in educational statistics is 5% or less, then getting 5 heads or five tails with 5 tosses would be insufficient for statistical significance unless you already suspected that the coin was a double-headed (tailed) penny, see one and two-tail test discussion below.

b) when a coin is tossed 6 times:

From row 6, of Pascal's triangle, it is evident that the probability of getting:

all heads is 1 in 64 1.5625%

all heads or all tails is 2 in 64 3.125%

Both of which are less than 5%; thus a minimum of 6 sets of nominal (head count) data is necessary to have any proof that the differences observed are unlikely to be due to chance factors. And yet, if 20 plus students repeating this experiment 5 times, it is probable that one obtain 6 heads or 6 tails. Probability is not about certainty!

II Assessing Probability in Practice

Although it is possible to calculate probability mathematically, tables to assess the probability are available in many statistics books. Simply look up the frequency of say X heads when a coin has been tossed T times. Normally Sign and Binomial tests are only done in place of one way Chi-square tests when the total number involved is equal to, or less than, 20.

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