Introduction
A measure of spread, sometimes likewise called a measure up of dispersion, is used to describe the variability in a sample or population. That is usually offered in conjunction with a measure up of main tendency, such as the typical or median, to carry out an as whole description that a collection of data.
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Why is it necessary to measure the spread out of data?
There are numerous reasons why the measure up of the spread of data values is important, yet one that the main reasons regards its relationship with measures of central tendency. A measure up of spread gives us an idea of how well the mean, because that example, represents the data. If the spread out of values in the data collection is large, the mean is not as representative that the data together if the spread of data is small. This is due to the fact that a huge spread indicates that there room probably huge differences in between individual scores. Additionally, in research, the is frequently seen as optimistic if there is little variation in each data group as it indicates that the similar.
We will be looking at the range, quartiles, variance, absolute deviation and standard deviation.
Range
The selection is the difference between the highest and lowest scores in a data collection and is the most basic measure the spread. So us calculate range as:
Range = maximum worth - minimum value
For example, let us consider the complying with data set:
23 | 56 | 45 | 65 | 59 | 55 | 62 | 54 | 85 | 25 |
The maximum worth is 85 and also the minimum worth is 23. This results in a selection of 62, which is 85 minus 23. Whilst using the range as a measure up of spread is limited, the does set the boundaries of the scores. This deserve to be beneficial if you room measuring a variable that has actually either a crucial low or high threshold (or both) that must not be crossed. The selection will instantly notify you whether at least one value broke these an important thresholds. In addition, the variety can be provided to detect any errors as soon as entering data. Because that example, if you have recorded the age of school youngsters in your study and your range is 7 come 123 years old you know you have made a mistake!
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Quartiles and also Interquartile Range
Quartiles tell us about the spread of a data set by breaking the data set into quarters, just like the median breaks that in half. Because that example, think about the marks of the 100 student below, which have actually been ordered indigenous the lowest to the highest possible scores, and also the quartiles highlighted in red.
order | Score | bespeak | Score | bespeak | Score | stimulate | Score | bespeak | Score |
1st | 35 | 21st | 42 | 41st | 53 | 61st | 64 | 81st | 74 |
2nd | 37 | 22nd | 42 | 42nd | 53 | 62nd | 64 | 82nd | 74 |
third | 37 | 23rd | 44 | 43rd | 54 | 63rd | 65 | 83rd | 74 |
4th | 38 | 24th | 44 | 44th | 55 | 64th | 66 | 84th | 75 |
fifth | 39 | 25th | 45 | 45th | 55 | 65th | 67 | 85th | 75 |
6th | 39 | 26th | 45 | 46th | 56 | 66th | 67 | 86th | 76 |
7th | 39 | 27th | 45 | 47th | 57 | 67th | 67 | 87th | 77 |
8th | 39 | 28th | 45 | 48th | 57 | 68th | 67 | 88th | 77 |
nine | 39 | 29th | 47 | 49th | 58 | 69th | 68 | 89th | 79 |
10th | 40 | 30th | 48 | 50th | 58 | 70th | 69 | 90th | 80 |
11th | 40 | 31st | 49 | 51st | 59 | 71st | 69 | 91st | 81 |
12th | 40 | 32nd | 49 | 52nd | 60 | 72nd | 69 | 92nd | 81 |
13th | 40 | 33rd | 49 | 53rd | 61 | 73rd | 70 | 93rd | 81 |
14th | 40 | 34th | 49 | 54th | 62 | 74th | 70 | 94th | 81 |
15th | 40 | 35th | 51 | 55th | 62 | 75th | 71 | 95th | 81 |
16th | 41 | 36th | 51 | 56th | 62 | 76th | 71 | 96th | 81 |
17th | 41 | 37th | 51 | 57th | 63 | 77th | 71 | 97th | 83 |
18th | 42 | 38th | 51 | 58th | 63 | 78th | 72 | 98th | 84 |
19th | 42 | 39th | 52 | 59th | 64 | 79th | 74 | 99th | 84 |
20th | 42 | 40th | 52 | 60th | 64 | 80th | 74 | 100th | 85 |
The first quartile (Q1) lies between the 25th and also 26th student"s marks, the second quartile (Q2) in between the 50th and also 51st student"s marks, and also the third quartile (Q3) between the 75th and also 76th student"s marks. Hence:
First quartile (Q1) = (45 + 45) ÷ 2 = 45Second quartile (Q2) = (58 + 59) ÷ 2 = 58.5Third quartile (Q3) = (71 + 71) ÷ 2 = 71
In the above example, we have an even variety of scores (100 students, fairly than one odd number, such as 99 students). This way that as soon as we calculation the quartiles, we take the sum of the two scores about each quartile and also then fifty percent them (hence Q1= (45 + 45) ÷ 2 = 45) . However, if we had an odd number of scores (say, 99 students), we would only have to take one score because that each quartile (that is, the 25th, 50th and 75th scores). You must recognize that the second quartile is likewise the median.

Quartiles space a useful measure the spread because they are much less affected by outliers or a it was crooked data set than the tantamount measures the mean and also standard deviation. For this reason, quartiles are regularly reported together with the mean as the best selection of measure up of spread out and main tendency, respectively, when handling skewed and/or data through outliers. A common method of to express quartiles is together an interquartile range. The interquartile variety describes the difference in between the third quartile (Q3) and also the an initial quartile (Q1), telling us around the range of the middle fifty percent of the scores in the distribution. Hence, because that our 100 students:
Interquartile selection = Q3 - Q1= 71 - 45= 26
However, it need to be noted that in journals and also other publications you will normally see the interquartile selection reported together 45 come 71, quite than the calculated range.
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A slim variation top top this is the semi-interquartile range, which is half the interquartile variety = ½ (Q3 - Q1). Hence, for our 100 students, this would certainly be 26 ÷ 2 = 13.
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