Statistical analysis

Learning Objectives Success criteria To understand the variation in living organisms Success criteria Learners should be able to demonstrate and apply their knowledge and understanding of: (f) the different types of variation An opportunity to use standard deviation to measure the spread of a set of data and/or Student’s t-test to compare means of data values of two populations the Spearman’s rank correlation coefficient to consider the relationship of the data.

Are our results reliable enough to support a conclusion?

Imagine we chose two children at random from two class rooms… … and compare their height …

… we find that one pupil is taller than the other C1 … we find that one pupil is taller than the other WHY?

REASON 1: There is a significant difference between the two groups, so pupils in C1 are taller than pupils in D8 D8 C1 YEAR 7 YEAR 11

REASON 2: By chance, we picked a short pupil from D8 and a tall one from C1 D8 C1 TITCH (Year 9) HAGRID (Year 9)

How do we decide which reason is most likely? MEASURE MORE STUDENTS!!!

… the average or mean height of the two groups should be very… If there is a significant difference between the two groups… D8 C1 … the average or mean height of the two groups should be very… … DIFFERENT

… the average or mean height of the two groups should be very… If there is no significant difference between the two groups… D8 C1 … the average or mean height of the two groups should be very… … SIMILAR

Living things normally show a lot of variation, so… Remember:

It is VERY unlikely that the mean height of our two samples will be exactly the same C1 Sample Average height = 162 cm D8 Sample Average height = 168 cm Is the difference in average height of the samples large enough to be significant?

Here, the ranges of the two samples have a small overlap, so… We can analyse the spread of the heights of the students in the samples by drawing histograms 2 4 6 8 10 12 14 16 Frequency 140-149 150-159 160-169 170-179 180-189 Height (cm) C1 Sample D8 Sample Here, the ranges of the two samples have a small overlap, so… … the difference between the means of the two samples IS probably significant.

Here, the ranges of the two samples have a large overlap, so… 2 4 6 8 10 12 14 16 Frequency 140-149 150-159 160-169 170-179 180-189 Height (cm) C1 Sample Here, the ranges of the two samples have a large overlap, so… … the difference between the two samples may NOT be significant. 2 4 6 8 10 12 14 16 Frequency 140-149 150-159 160-169 170-179 180-189 Height (cm) D8 Sample The difference in means is possibly due to random sampling error

Calculators and computer programs will do this for you!!!!!!!!!!! To decide if there is a significant difference between two samples we must compare the mean height for each sample… … and the spread of heights in each sample. Statisticians calculate the standard deviation of a sample as a measure of the spread of a sample Sx = Σx2 - (Σx)2 n n - 1 Where: Sx is the standard deviation of sample Σ stands for ‘sum of’ x stands for the individual measurements in the sample n is the number of individuals in the sample You can calculate standard deviation using the formula: Calculators and computer programs will do this for you!!!!!!!!!!!

You DO need to know the concept! Standard deviation is a statistic that tells how tightly all the various datapoints are clustered around the mean in a set of data. When the data points are tightly bunched together and the bell-shaped curve is steep, the standard deviation is small.(precise results, smaller sd). This could indicate greater reliability. When the data points are spread apart and the bell curve is relatively flat, a large standard deviation value suggests less precise results. This may indicate lower reliability.

Calculators and computer programs will do this for you!!!!!!!!!!! Student’s t-test The Student’s t-test compares the averages and standard deviations of two samples to see if there is a significant difference between them. We start by calculating a number, t t can be calculated using the equation: ( x1 – x2 ) (s1)2 n1 (s2)2 n2 + t = Where: x1 is the mean of sample 1 s1 is the standard deviation of sample 1 n1 is the number of individuals in sample 1 x2 is the mean of sample 2 s2 is the standard deviation of sample 2 n2 is the number of individuals in sample 2 Calculators and computer programs will do this for you!!!!!!!!!!!

What is the t-test? The t-test is used to test the difference between two sets of data. It provides a way of measuring the overlap between the two sets of data. First we state a null hypothesis that there is no significant difference between the means of 2 sets of data. If the two sets of data show widely separated means and small variances, they will have little overlap and a big value of t If the two sets of data have means that are close together and large variances, they will have a lot of overlap and a small value of t

Certainly or probably? Large t = almost ‘certainly’- that is- unlikely to be due to chance. Small t = ‘probably’ no difference. If t is greater than or equal to the critical value (from t-tables)then there is a significant difference between the two sets of data. The degrees of freedom is the number samples in each group –2 i.e.(n1 + n2 –2)

Correlation coefficient This is used to consider the relationships between 2 sets of data. The Spearman Rank correlation tells us whether 2 sets of data are correlated or not. rs = 1 - 6 ΣD2 n(n2 – 1) Where rs is the rank coefficient D is the difference between the ranks N is the number of pairs of values Again we need a null hypothesis and we compare our answer to a table of critical values.