Facts from figures Having obtained the results of an investigation, a scientist is faced with the prospect of trying to interpret them. In some cases the results may be clear-cut, but generally some form of statistical analysis is necessary.

Population mean vs. sample mean Mean: the quotient of the sum of several quantities and their number; the average.

Population mean vs. sample mean Population mean: the value obtained by taking measurements from every individual in a population.

Population mean vs. sample mean Sample mean: the value obtained by taking measurements from some individuals in a population.

Which is more reliable? Sample statistics are subject to error; population statistics are “true” (i.e. in accordance with fact or reality). Sampling is used to estimate population statistics, and it must be remembered that sampling is always prone to error.

If we have two independent samples, how can we tell if an apparent difference between the means is real, or simply due to chance?

In this case the means are compared using a t-test In this case the means are compared using a t-test. The difficulty is that because of chance (the absence of design or discoverable cause), the means of any two samples would be expected to differ.

A difference between the sample mean and the population mean is due to sampling error (an effect that is purely due to chance).

Just how much can two sample means differ before we can reasonably conclude that they were taken from populations that, as far as the variable that was measured is concerned, are significantly different?

A worked example Table showing the number of red blood cells in each of the randomly selected squares.

Sample 1 Sample 2 2 3 6 4 5 1 8

Null hypothesis The t-test literally tests how likely it is that there is no real difference between the means of the two samples; i.e. it is testing a null hypothesis.

Null hypothesis There is no significant difference between the mean number of red blood cells in sample one and sample two.

The t- test 1. clear memory 2. go to stats mode 3. enter data 4. press mean; record 5. press standard deviation: σ n-1

The t- test 6. square this to get the variance 7. divide by n (sample size), then take the square root to get the standard error 8. square this and enter this value in the equation

The t- test 9. enter into formula for t-test 10. clear memory and repeat for second sample 11. complete calculation to obtain the t value

σ n-1 (standard deviation) = 1.95 Answer: sample 1 mean = 4.7 σ n-1 (standard deviation) = 1.95 variance = 3.8 standard error = 0.62 square this = 0.38 and enter this in the equation

σ n-1 (standard deviation) = 1.25 Answer: sample 2 mean = 2.3 σ n-1 (standard deviation) = 1.25 variance = 1.56 standard error = 0.395 square this = 0.16 and enter this in the equation

Answer 4.7 - 2.3 t = 0.38 + 0.16 = 3.3

The t- test Turn to the table for t-tests: you will only need one row, for the degrees of freedom: n1 + n2 -2

The t- test In this example n1 + n2 –2 = 18

The t- test Read along the row and find the t value nearest yours. Describe where your value lies, using the p values at the top of the table.

t = 3.3

t = 3.3 d.f. = 10 + 10 –2 = 18 p>0.002, and p<0.01 therefore:

Since p is less than 0.05, you MUST REJECT the null hypothesis. This is because the chance of the null hypothesis being true is very small, less than 5%.

Conclusion: there is a significant difference between the mean number of RBCs in the two samples; it is greater in sample one.

However if p had been greater than 0 However if p had been greater than 0.05, you MUST ACCEPT the null hypothesis. This is because the chance of the null hypothesis being true is relatively high, greater than the cut-off value of 5%.

Conclusion: there is no significant difference between the mean number of RBCs in the two samples; any differences are due to sampling error.

p value conclusion p<0.05 p>0.05 decision re: null hypothesis REJECT e.g. mean of sample 1 is significantly greater than the mean of sample 2 p>0.05 ACCEPT no significant difference between the means of the two samples