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# The normal distribution

Extracts from this document...

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 deviations 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 dataSubtract each score from the meanTimes each difference by itself (negs turn positive)Add up all the squared differencesDivide the total by the number of scores minus 1Take the square root

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 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

1. Put the values into the equation and work it out carefully!
2. Note down the value of t found. In this case it is 3.08.
3. 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.
4. Find the value of t

Conclusion

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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

1. 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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