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Understanding Data Spread

Measures of variation and spread describe how dispersed or spread out values in a data set are. They are typically used alongside measures of central tendency like the mean or median. The range is the simplest measure of spread, being the difference between the highest and lowest values. Quartiles divide data into quarters to measure spread in a way less affected by outliers. The interquartile range describes the middle half of scores. Standard deviation quantifies how far values are from the mean, with larger standard deviations indicating greater variation. It is calculated by taking the square root of the variance, which is the average of squared differences from the mean.

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518 views5 pages

Understanding Data Spread

Measures of variation and spread describe how dispersed or spread out values in a data set are. They are typically used alongside measures of central tendency like the mean or median. The range is the simplest measure of spread, being the difference between the highest and lowest values. Quartiles divide data into quarters to measure spread in a way less affected by outliers. The interquartile range describes the middle half of scores. Standard deviation quantifies how far values are from the mean, with larger standard deviations indicating greater variation. It is calculated by taking the square root of the variance, which is the average of squared differences from the mean.

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Measures of Variation/Spread

Definition
A measure of spread, sometimes also called a measure
of dispersion, is used to describe the variability in a
sample or population. It is usually used alongside
a measure of central tendency, such as the mean or
median, to provide an overall description of a set of
data.

Why calculate spread?


A measure of spread gives us an idea of how well the
mean represents the data. If the spread is large the
mean isnt a good representation of the data. This is
because a large spread indicates that there are
probably large differences between individual scores.

Range
The range is the difference between the highest and
lowest scores in a data set and is the simplest measure
of spread. So we calculate range as:

Range = maximum value - minimum value

Quartiles and Interquartile Range


Quartiles tell us about the spread of a data set by
breaking the data set into quarters, just like the median
breaks it in half. Quartiles are a useful measure of
spread because they are affected less by outliers. The
interquartile range describes the difference between
the third quartile (Q3) and the first quartile (Q1), telling
us about the range of the middle half of the scores in
the distribution.

Standard Deviation
The Standard Deviation is a measure of how spread out
numbers are.

Its symbol is (the Greek letter sigma)

The formula is easy: it is the square root of


the Variance.

Variance
The Variance is defined as:

The average of the squared differences from the


Mean.

To calculate the variance follow these steps:

-Work out the Mean (the simple average of the


numbers)
-Then for each number: subtract the Mean and square
the result (the squared difference).
-Then work out the average of those squared
differences.

Example
You and your friends have just measured the heights of
your dogs (in millimetres):

The heights (at the shoulders) are: 600mm, 470mm,


170mm, 430mm and 300mm.

Find out the Mean, the Variance, and the Standard


Deviation.

Your first step is to find the Mean:

Answer:
Mean = 600 + 470 + 170 + 430 + 3005 = 19705 =
394

so the mean (average) height is 394 mm.

Now we calculate each dog's difference from the Mean:

To calculate the Variance, take each difference, square


it, and then average the result:
So the Variance is 21,704

And the Standard Deviation is just the square root of


Variance, so:

Standard Deviation

= 21,704
= 147.32...
= 147 (to the nearest mm)

And the good thing about the Standard Deviation is


that it is useful. Now we can show which heights are
within one Standard Deviation (147mm) of the Mean:

So, using the Standard Deviation we have a "standard"


way of knowing what is normal, and what is extra-large
or extra small.

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