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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

  1. Subtract each score from the mean
  2. Times each difference by itself (negs turn positive)
  3. Add up all the squared differences
  4. Divide the total by the number of scores minus 1
  5. Take the square root
...read more.

Middle

image07.png

image08.png

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 (image09.png)

39.93

34.93

Standard deviation (s)

2.73

3.31

Number of scores (n)

7

7

image11.pngimage01.pngimage02.pngimage10.png

  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
...read more.

Conclusion

png" style="width:648px;height:469.47px;margin-left:0px;margin-top:0px;" alt="image03.png" />

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

...read more.

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