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Quantitative data collection and analysis

Proababilty, Proportions and Odds

A probability refers to the likelihood of an event occurring.

With statistics we are attempting to infer from a sample the probability that the null hypothesis is true or false.


Sometimes you may be more interested in proportions rather than Means.

e.g. the proportion of teenagers who have been excluded from mainstream education. 

To work out the proportion divide the number of instances you are interested in by the total number in your sample.

(to make it a percentage times this result by 100).


Z scores

As is the case for sample means if we take repeated samples and find the proportion p of each of these samples then the mean of all these samples should approximate the population proportion.

The Z score allows you to estimate the chance of an outcome occurring.

p = the sample proportion

π = (‘pie’) population proportion

= number in sample


Confidence intervals can be used when working with proportions to ascertain whether your sample proportion reflects what would be found in the population, however the calculation is different.

e.g. taken from Foster, L., Diamond, I. and Jefferies, J. (2015) Beginning statistics: an introduction for Social Scientists. 2nd edn. London: SAGE, p.152.

Out of 200 pensioners surveyed. 58 replied that they had retired when they expected to.

That is p = 29% or 0.29 

We can conclude with 95% confidence that the proportion of pensioners in the population who retired when expected would be between 0.2271 and 0.3529 i.e. between 23% and 35% of pensioners.

Odds can be considered an alternative statistic to proportions.

The observed odds of a response is:

The number of people with that response divided by all the people without that response.

This gives you a proportion:

If the proportion in the sample is 0.50 (i.e. half or 50% of the sample) the odds will be be 1.00.

If the proportion in the sample is greater than 0.50 then the odds will be more than 1.00

If the proportion in the sample is less than 0.50 then the odds will be less than 1.00.