2. Chi Square Analysis  Use to see if the observed value varies from the expected value.  Null Hypothesis – There is no difference between the observed and expected – any difference is due to chance or sampling error  If there is > 5% chance that the difference is due to random error then we accept the Null hypothesis meaning there is no difference.

3. Example  If you flip a coin – you expect 50/50 heads and tails. If you get something different – is it just due to random chance or is it a real difference – maybe due to a faulty coin?  Calculating Chi Square:  Chi-squared = �  (observed-expected) 2 /(expected)  We have two classes to consider in this example, heads and tails.  Chi-squared = (100-108) 2 /100 + (100-92) 2 /100 = (- 8) 2 /100 + (8) 2 /100 = 0.64 + 0.64 = 1.28 Heads TailsTotal Observed10892200 Expected100 200 Total208192400

4. How to use Chi Square to tell if there is a difference df/pro b. 0.990.950.900.800.700.500.300.200.100.05 1 0.0001 3 0.00390.0160.640.150.461.071.642.713.84 20.020.100.210.450.711.392.413.224.605.99 30.120.350.581.001.422.373.664.646.257.82 40.30.711.061.652.203.364.885.997.789.49 50.551.141.612.343.004.356.067.299.2411.07 DF = degrees of freedom = # of categories -1 (You have to take into account the # of categories because the more there are the more deviation you would expect DF = 1 (2 categories heads & tails -1) Using 1DF – look up 1.28 on the chart = ~0.27 which means that 27% of the time this would happen due to pure chance - there is a 27% chance that it’s not different from the expected – or there is a 27% chance that it is a “fair coin” – or there is a 73% chance that it is biased – however anything > 5% chance that it is due to random sampling is accepted – therefore we accept our Null hypothesis that the coin is fair and that 108/200 heads is the same as 100/200

5. Example #2  Two tigers have four cubs – two are albino  You hypothesize that both parents are heterozygous  If they are Oo, what is the probability of having an albino baby – so how many orange and how many white cubs out of 4?  Calculate Chi Square and see if you accept the Null Hypothesis which says the difference you see is due to random sampling error.

6. Answer Therefore – it is > 20% probability that the difference between what we saw (2 albino cubs) and what we expected (1 albino cub) is due to chance alone. Remember – if it > 5% we accept the Null Hypothesis and conclude they aren’t different

7. Standard Deviation  Standard deviation (SD) (represented by the Greek letter sigma, σ) is used to quantify the amount of variation. A standard deviation close to 0 indicates that the data points tend to be very close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.σmean A plot of a normal distribution (or bell- shaped curve) where each band has a width of 1 standard normal distribution

8. How to use and calculate SD  The standard deviation is found by taking the square root of the average of the squared deviations of the values from their average value. See the formula sheet.square rootaverage  For example, the scores of eight students randomly selected from a class of 25 students are shown below:  2,4,4,4,5,5,7,9 – mean = 5 What is the SD for these scores?  Square each difference from the mean  Add these squared differences  Divide by n-1 (# of sample size-1)  Then take the square root of that Note- if you data from the entire population and not a sample – you would divide by n and not n-1