MEASURES OF SPREAD
MEASURES OF SPREAD
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STATISTICAL MEASURES
The following sets of data are test results obtained by a group of students in 2 tests in which the maximum mark was 20:
TEST 1 (place chart here)
TEST 2 (place chart here)
The means of the 2 tests are fairly close to each other (test1=10.1, test2=9.5); however, there is a substantial difference between the 2 sets which can be seen from the frequency tables:
Test 1 frequencies:
MARK---------------FREQUENCY
3----------------------------2-------------------
4-----------------------------2------------------
5-----------------------------2-----------------
6-----------------------------4-----------------
7-----------------------------3-----------------
8-----------------------------2-----------------
9---------------------------- 1-----------------
10-------------------------- 8-----------------
11-------------------------- 5-----------------
12-------------------------- 8-----------------
13-------------------------- 6-----------------
14-------------------------- 2-----------------
15-------------------------- 0-----------------
16---------------------------1-----------------
17---------------------------0-----------------
18---------------------------0-----------------
19---------------------------2-----------------
Test 2 Frequencies:
MARK------------------FREQUENCY
3----------------------------0
4----------------------------0
5----------------------------0
6----------------------------0
7-----------------------------0
8-----------------------------13
9------------------------------11
10-----------------------------12
11-----------------------------12
12------------------------------0
13-------------------------------0
14-------------------------------0
15------------------------------0
16-------------------------------0
17-------------------------------0
18-----------------------------0
19----------------------------0
The marks for test 1 are very spread out across the available scores whereas those for test 2 are concentrated around 9,10, and 11. This may be important as the usual reason for setting tests is to rank the students in order of their performance. test 2 is less effective at this than test 1 because the marks have a very small spread. In fact, when teachers/examiners set a test, they are more interested in getting a good spread of marks than they are in getting a particular value for the mean. By contrast, manufacturers of precision engineering products want a small spread on the dimensions of the articles that they make. either way, it is necessary to have a way of calculating a numerical measure of the spread of data. the most commonly used measures are variance, standard deviation, and interquartile range.
VARIANCE AND STANDARD DEVIATION:
~To calculate the variance of a set of data, the frequency table can be extended as follows:
TEST 1:
MARK---------------FREQUENCY-----------M- µ---------------f(M- µ)^2
3----------------------------2------------------- -7.10------------- 100.82
4-----------------------------2------------------ -6.10------------- 74.42
5-----------------------------2----------------- -5.10------------- 52.02
6-----------------------------4----------------- -4.10------------- 67.24
7-----------------------------3----------------- -3.10------------- 28.83
8-----------------------------2----------------- -2.10------------- 8.82
9---------------------------- 1----------------- -1.10------------- 1.21
10-------------------------- 8----------------- -0.10------------- 0.08
11-------------------------- 5----------------- 0.90------------- 4.05
12-------------------------- 8----------------- 1.90------------- 28.88
13-------------------------- 6----------------- 2.90------------- 50.46
14-------------------------- 2----------------- 3.90------------- 30.42
15-------------------------- 0----------------- 4.90------------- 0.00
16---------------------------1----------------- 5.90------------- 34.81
17---------------------------0----------------- 6.90------------- 0.00
18---------------------------0----------------- 7.90------------- 0.00
19---------------------------2----------------- 8.90-------------158.92
-------------------------------------------------------------TOTAL: 640.48
The third column in this table measures the amount that each mark deviates from the mean mark of 10.10. Because some of these marks are larger than the mean and some are smaller, some of these deviations are opsitive and some are negative. If we try to calculate an average deviation using these results, the negative deviations wil cancel out the positive deviations. to correct this problem, one method is to square the deviation. Finally, this result is multiplied by the frequency to produce the results in the fourth column. in detail, the last row is calculated 2 x (3 -- 10.10)^2 = 2 x 50.41 = 100.82. the total of the fourth column is divided by the number of data items (48) to obtain the variance of the marks:
VARIANCE = 640.48 / 48 = 13.34.
The measure most commonly used is the square root of the variance (remember that we squared the deviations). A measure is used that is known as the standard deviation of the marks. in the previous case:
-----------------Standard Deviation = (13.34)^.5 = 3.65.
Repeating this calculation for the second set of marks:
MARK (M)----------------------FREQUENCY----------------- (M -- m) --------------- f (M -- m)^2
8---------------------------------------------13-------------------------- --1.48---------------------28.475
9----------------------------------------------11------------------------ --0.48-----------------------2.534
10-------------------------------------------12-----------------------------0.52-----------------------3.245
11--------------------------------------------12----------------------------1.52-----------------------27.725
---------------------------------------------------------------------------------------------TOTAL: 61.979
VARIANCE = 61.979 / 48 = 1.291
STANDARD DEVIATION: (1.291)^.5 = 1.136
This figure is about one third of the figure calculated for Test 1. This reflects the fact that Test 2 has not spread the students very well. In summary, the variance and population standard deviation are calculated using the formulas:
*****VARIANCE = (greek formula here p 219)
*****POPULATION STANDARD DEVIATION = (greek formula here p219)
USING A GRAPHING CALCULATOR:
