3.3 Measures of Spread

Minds On….

Describe the similarities and differences between the following two graphs:

- they both have 16 elements- mean, median, and mode are all 6- the data is more spread out on the left- the data on the right graph is more tightly clustered around the mean.

In the previous section we learned how to describe a set of data using measures of central tendency.

The terms spread, dispersion, & variation all refer to a measure of the way a data set is distributed about some central value. These measures will give us another numerical value that will allow us to describe a data set.

Just as there are several measures of central tendency, there are also different measures of spread.

- Interquartile Ranges- Standard Deviation- Percentiles

Part 1: Interquartile Ranges

A box and whisker plot is a visual representation of data divided into four groups (quartiles) with equal numbers of values in each quartile.

The three dividing points are:

Q1 - median of lower half of dataQ2 - median of all dataQ3 - median of upper half of data

The Interquartile Range is the range of the middle half of the data ( = Q3 - Q1 ). The larger the interquartile range, the larger the spread of the central half of the data. This is a good measure of spread (dispersion) but can be tedious to calculate.

Remember: The median is the middle value in an ordered data set. If the set has an even number of data points, then the median is halfway between the two middle-most values.

Example 1

Here are the final grades in a grade 9 physical education class:

88, 56, 72, 67, 59, 48, 81, 6290, 75, 75, 43, 71, 64, 78, 84

a) First put the data into numerical order

This is always the first step when determining the quartiles

43, 48, 56, 59, 62, 64, 67, 71, 72, 75, 75, 78, 81, 84, 88, 90

b) Determine Q1, Q2, and Q3

Range =

Median = Q2 =

Q1 =

Q3 =

IQR =

Note: although a quartile is, strictly speaking, a single value, people sometimes speak of datum being within a quartile. What they really mean is that the datum is in the quarter whose upper boundary is the quartile. Example: if a value x is within the second quartile, then Q1 < x ≤ Q2

d) If your final mark was 75%, what quartile were you within?

Example 2a) Calculate the IQR for each company. What does it tell you?

Wait times for a pizza:

Q1 - 8.4 minutes

Q2 - 14.6 minutes

Q3 - 16.3 minutes

IQR = Q3-Q1

= 16.3 - 8.4

= 7.9 minutes

- median wait time is lower, so it appears to be faster

Wait times for a pizza:

Q1 - 14.4 minutes

Q2 - 16.2 minutes

Q3 - 17.3 minutes

IQR = Q3-Q1

= 17.3 - 14.4

= 2.9 minutes

- wait times are more consistent since the IQR is lower

b) If you wait 15 minutes for a pizza, what quartile are you within if you ordered from:

i) pizza pizza

ii) pizza hut

Part 2: Standard Deviation

The deviation of a piece of data is the distance it is from the mean of the set of data. If you were to take the all the deviations for an entire set of data, square each one of them, and then find the average, you would have what is called the variance. The square root of the variance is called the standard deviation.

The Greek letter sigma is used to represent variance and standard deviation

Standard Deviation Formula (Ungrouped Data)

So what does Standard Deviation tell us?

- if most data is clustered around the mean, the SD will be small (more consistent)

- if the data is spread out, the SD will be large (less consistent)

larger is more spread

smaller is more compact

Example 3

Here is a list of a students grade 12 course marks:

80% 89% 75% 83%

a) Calculate the mean

b) Calculate the standard deviation

c) What does the standard deviation tell you?

The standard deviation tells you how dispersed the data is about the mean. The standard deviation is relatively small, therefore the data is clustered closely to the mean.

Example 4Felix and Melanie are laying down patio stones. They record how many stones they put in place each hour.

a) Which worker gets more done during the day?

b) Which worker is more consistent? (calculate the standard deviation for each worker and compare them)

Felix

Start by finding the mean:

Melanie

c) Who is a more consistent worker?

The standard deviation of Felix's set of numbers is smaller. This means his data are more closely clustered about the mean; in other words, he is a more consistent worker.

The standard deviation of Melanie's numbers is considerably larger, indicating that the data are more widely dispersed away from the mean. As a result, Melanie is less consistent.

Part 3: Standard Deviation (Grouped Data)

Formulas you need:

mean for weighted data

standard deviation for grouped data

A railway line gives out small bags of peanuts to its travellers, and each bag doesn't always contain the same number of peanuts. The following table represents a sample of 31 bags showing the number of peanuts per bag.

Example 5

a) Calculate the mean number of peanuts per bag.

b) Calculate the standard deviation for this sample

c) Calculate the standard deviation for this sample using technology.

page 168 #1, 2, 3bc (by hand) #3ad (with technology), 5 SD (by hand and calc),6, 7a (with calc), 9, 10