- The three averages (
**mean**,**median**and**mode**) measure what is called**central****tendency**- they all give an indication of what is
*typical*about the data - what lies
*roughly*in the middle

- they all give an indication of what is
- The
**range**and**inter****quartile****range**(**IQR**) measure how**spread**out the data is- These can only be applied to numerical data
- Fortunately both are easy to work out!

- The
**range**is the**difference**between the**highest data value**and the**lowest data value**- It measures how
**spread**out the data is - You can remember this as "Hi - Lo"

- It measures how

- There is one possible problem with the range
- as it considers the highest and lowest values in the data set it could be influenced by anomalies (outliers) in the data
- these may not be a true representation of how spread the rest of the data may be

- The median splits the data set into two parts, lying half way along the data
- As their name suggests, quartiles split the data set into four parts
- The
**lower quartile****(LQ)**lies a**quarter**of the way along the data (when in order) - The
**upper quartile****(UQ)**lies**three quarters**of the way along the data - You may come across the median being referred to as the second quartile

- The

- As their name suggests, quartiles split the data set into four parts
- To find the quartiles first use the median to divide the data set into lower and upper halves

- Make sure the data is put into numerical order first
- If there are an even number of data values, then the first half of those values are the lower half, and the second half are the upper half
- In this case, all the numbers in the data set are included in one or the other of the two halves

- If there are an odd number of data values, then all the values
**below**the median are the lower half and all the values**above**the median are the upper half- In this case, the median itself is not included as a part of either half

- The lower quartile is the median of the lower half of the data set, and the upper quartile is the median of the upper half of the data set
- Find the quartiles in the same way you would find the median for any other data set
- just restrict your attention to the lower or upper half of the data accordingly

- Find the quartiles in the same way you would find the median for any other data set
- Sometimes you may also see the quartiles given in formula form

- For
*n*data values:- the
**lower quartile**is the value - the
**upper quartile**is the value

- the
- Using these can save finding the median and splitting the data into two halves

- For

- This is the difference between the upper quartile (UQ) and the lower quartile (LQ)

- So you need to find the quartiles first before you can calculate the interquartile range

- The interquartile range is
**IQR = UQ - LQ** - The interquartile range considers the middle 50% of the data so is not affected by extreme values in the data
- The range of a data set, on the other hand, can be affected by extremely large or small values

- Remember with the range that you have to do a calculation (even if it is an easy subtraction)
- it is not enough to write something like the range is 14 to 22
- the same applies to the interquartile range

a)

Find the range and interquartile range for the following data.

3.4 | 4.2 | 2.8 | 3.6 | 9.2 | 3.1 | 2.9 | 3.4 | 3.2 |

3.5 | 3.7 | 3.6 | 3.2 | 3.1 | 2.9 | 4.1 | 3.6 | 3.8 |

3.4 | 3.2 | 4.0 | 3.7 | 3.6 | 2.8 | 3.9 | 3.1 | 3.0 |

The values need to be in order - work carefully to ensure you do not miss any out.

2.8 | 2.9 | 2.9 | 3.0 | 3.1 | 3.1 | 3.2 | ||

3.2 | 3.2 | 3.4 | 3.4 | 3.5 | 3.6 | 3.6 | 3.6 | |

3.6 | 3.7 | 3.8 | 3.9 | 4.0 | 4.1 | 4.2 |

Working out the range is now very easy.

Range = Hi - Lo = 9.2 - 2.8 = 6.4

To find the interquartile range, we first need to find the median and split the data into two halves.

There are 27 data values, so the median is the 14 ^{th} value (3.4).

The lower half of the data set is

2.8, 2.8, 2.9, 2.9, 3.0, 3.1, **3.1**, 3.1, 3.2, 3.2, 3.2, 3.4, 3.4 (all the values below the median)

There are 13 values, so the lower quartile is the 7 ^{th} value (3.1).

Remember, the LQ is the median of the lower half of the data set!

LQ = 3.1

The upper half of the data set is

3.5, 3.6, 3.6, 3.6, 3.6, 3.7, **3.7**, 3.8, 3.9, 4.0, 4.1, 4.2, 9.2 (all the values above the median)

There are 13 values, so the upper quartile is the 7 ^{th} value (3.7). (The 7th of *these* values.)

Remember, the UQ is the median of the upper half of the data set!

UQ = 3.7

Now subtract the LQ from the UQ to find the IQR.

IQR = UQ - LQ = 3.7 - 3.1 = 0.6

**The range is 6.4 The interquartile range is 0.6 **

Alternatively, use the formulae to locate the LQ and UQ.

The LQ is the value; LQ = 3.1

The UQ is the value; UQ = 3.7

b)

Give a reason why, in this case, the interquartile range may be a better measure of how spread out the data is than the range.

**The IQR would be a better measure of spread for these data as the highest value (9.2) is very far away from the rest of the numbers - it could be an outlier**

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