The normal distribution
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Introduction
The normal distribution
When many measures are taken of something (eg, scores in a test, people's heights, pollution levels in rivers) the spread of the values will have a bell shape, called the normal distribution.
A number of statistical tests use this characteristic distribution (or dispersion) of values to test whether two samples are the same or different.
There are several basic terms that are commonly used with the normal distribution.
Average (mean)  A measure of the average score in a set of data. The mean is found by adding up all the scores and then dividing by the number of scores. 
Range  The difference between the largest core and the and smallest score. 
Median  If a set of scores are arranged from lowest to highest the median is the score in the middle, with half above and half below. 
Mode  The value that occurs most often 
Standard deviation s  A measure of the standard (average) deviation of the scores from the mean.The larger the standard deviation the larger the range of values/variation in the data

Middle
Then calculate the square root to get the standard deviation
Comparing two samples: using the t test
The average, standard deviation and the number of scores in each sample are the three things needed to do a t test. A t test is used with two samples of data to test whether they are significantly different (ie, whether one is truly higher or lower than the other). The same sample of scores as used above is now compared with another sample of scores.
Sample 1 scores  Sample 2 scores  
41  38  
43  32  
37.5  35.5  
38.5  33  
44  31.5  
38  40.5  
37.5  34  
Average ()  39.93  34.93 
Standard deviation (s)  2.73  3.31 
Number of scores (n)  7  7 
 Put the values into the equation and work it out carefully!
 Note down the value of t found. In this case it is 3.08.
 You will also need to know how many degrees of freedom to use with the critical values of t table. Degrees of freedom = (nsample1 + nsample2) – 2 . In this example this equals 7 + 7 –2 = 12.
 Find the value of t
Conclusion
2. Calculate the average, range, median and mode for the following set of data (a random set of your exam results from the last exam): 66.25, 15, 32.5, 26.25, 48.75, 48.75, 36.25, 35, 68.75, 72.5, 43.75, 40, 20, 48.75, 12.5, 41.25, 53.75, 50, 31.25, 95, 22.5, 33.75, 27.5, 55, 12.5, 45, 18.75, 42.5, 62.5, 85, 75
Degrees of freedom  Value of t that must be exceeded (5% level) 
1  12.706 
2  4.303 
3  3.182 
4  2.776 
5  2.571 
6  2.447 
7  2.365 
8  2.306 
9  2.262 
10  2.228 
11  2.201 
12  2.179 
13  2.160 
14  2.145 
15  2.131 
16  2.120 
17  2.110 
18  2.101 
19  2.093 
20  2.086 
22  2.074 
24  2.064 
26  2.056 
28  2.048 
30  2.042 
40  2.021 
60  2.000 
120  1.980 
3. The two sets of data given below are resting heart rates for a group of students and a group of professional athletes. Use the t test to find out if they are significantly different (using the table at right to test the value of t with the appropriate number of degrees of freedom). I need to see how the mean, standard deviation and t value were calculated.
Professional
Students athletes
57.1 61.7
47.6 47.0
58.0 55.5
74.8 62.6
 41.8
51.9 60.8
64.2 50.2
49.6 44.2
67.2 45.4
62.6 39.3
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