2Are our results reliable enough to support a conclusion?
3Imagine we chose two children at random from two class rooms… … and compare their height …
4… we find that one pupil is taller than the other C1… we find that one pupil is taller than the otherWHY?
5REASON 1:There is a significant difference between the two groups, so pupils in C1 are taller than pupils in D8D8C1YEAR 7YEAR 11
6REASON 2:By chance, we picked a short pupil from D8 and a tall one from C1D8C1TITCH(Year 9)HAGRID(Year 9)
7How do we decide which reason is most likely? MEASURE MORE STUDENTS!!!
8… the average or mean height of the two groups should be very… If there is a significant difference between the two groups…D8C1… the average or mean height of the two groups should be very…… DIFFERENT
9… the average or mean height of the two groups should be very… If there is no significant difference between the two groups…D8C1… the average or mean height of the two groups should be very…… SIMILAR
10Living things normally show a lot of variation, so… Remember:
11It is VERY unlikely that the mean height of our two samples will be exactly the same C1 SampleAverage height = 162 cmD8 SampleAverage height = 168 cmIs the difference in average height of the samples large enough to be significant?
12Here, 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 histograms246810121416FrequencyHeight (cm)C1 SampleD8 SampleHere, the ranges of the two samples have a small overlap, so…… the difference between the means of the two samples IS probably significant.
13Here, the ranges of the two samples have a large overlap, so… 246810121416FrequencyHeight (cm)C1 SampleHere, the ranges of the two samples have a large overlap, so…… the difference between the two samples may NOT be significant.246810121416FrequencyHeight (cm)D8 SampleThe difference in means is possibly due to random sampling error
14… and the spread of heights in each sample. 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 sampleSx =Σx2 -(Σx)2nn - 1Where:Sx is the standard deviation of sampleΣ stands for ‘sum of’x stands for the individual measurements inthe samplen is the number of individuals in the sampleYou can calculate standard deviation using the formula:
15It is much easier to use the statistics functions on a scientific calculator! e.g. for data 25, 34, 13Set calculator on statistics modeMODE2(CASIO fx-85MS)Clear statistics memorySHIFTCLR1(Scl)=Enter data25DT(M+ Button)341
16Calculate the standard deviation Calculate the mean24ACSHIFTS-VAR(2 Button)1( x )=Calculate the standard deviationACSHIFTS-VAR3=(xσn-1)
17We start by calculating a number, t Student’s t-testThe 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, tt can be calculated using the equation:( x1 – x2 )(s1)2n1(s2)2n2+t =Where:x1 is the mean of sample 1s1 is the standard deviation of sample 1n1 is the number of individuals in sample 1x2 is the mean of sample 2s2 is the standard deviation of sample 2n2 is the number of individuals in sample 2
18Worked Example: Random samples were taken of pupils in C1 and D8 Their recorded heights are shown below…Students in C1Students in D8Student Height (cm)145149152153154148157161162158160166163167172175177182183185187Step 1: Work out the mean height for each sampleC1: x1 =161.60D8: x2 =168.27Step 2: Work out the difference in meansx2 – x1 = – =6.67
19Step 3: Work out the standard deviation for each sample C1: s1 =10.86D8: s2 =11.74Step 4: Calculate s2/n for each sample(s1)2n1=C1:÷ 15 =7.86(s2)2n2=D8:÷ 15 =9.19
21Step 7: Work out the number of degrees of freedom d.f. = n1 + n2 – 2 =– 2 =28Step 8: Find the critical value of t for the relevant number of degrees of freedomUse the 95% (p=0.05) confidence limitCritical value = 2.048Our calculated value of t is below the critical value for 28d.f., therefore, there is no significant difference between the height of students in samples from C1 and D8
22Follow the instructions CAREFULLY Do not worry if you do not understand how or why the test worksFollow the instructions CAREFULLYYou will NOT need to remember how to do this for your exam