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:

23564565595562548525

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.


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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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