(6.4) Measuring the Spread of Data - Discrete Data
We learned that the range and interquartile range were measures of the spread of data.
However, these two measures only use two values in their calculations and can change greatly even with small changes to the data.
For example,
Range =
Range =
However, these two measures only use two values in their calculations and can change greatly even with small changes to the data.
For example,
Range =
Range =
Variance and Standard Deviation
Variance and Standard deviation are also measures of the spread of data.Their calculations use all of the data values and how far they are from the mean.
For example,
Mean, $\mu$ =
| Data $x$ |
$5$ | $6$ | $6$ | $6$ | $6$ | $6$ | $7$ | $7$ | $7$ | $7$ | $7$ | $7$ | $7$ | $7$ | $7$ | $8$ | $8$ | $8$ | $8$ | $8$ | $9$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $x-\mu$ | $-2$ |
| $\left(x-\mu\right)^2$ | $4$ |
|---|
The sum of all $\left(x-\mu\right)^2 = \sum \left(x-\mu\right)^2 =$
The total number of data values, $n =$
Variance, $\sigma^2 = \frac{\sum \left(x-\mu\right)^2}{n} = $
*Give your answer to 3 significant figures
*Give your answer to 3 significant figures
Standard deviation, $\sigma = \sqrt{\frac{\sum \left(x-\mu\right)^2}{n}} = $
*Give your answer to 3 significant figures
*Give your answer to 3 significant figures
Example 2
Mean, $\mu$ =
Mean, $\mu$ =
| Data $x$ |
$4$ | $4$ | $5$ | $5$ | $5$ | $6$ | $6$ | $7$ | $7$ | $7$ | $7$ | $7$ | $7$ | $8$ | $8$ | $9$ | $9$ | $9$ | $10$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $x-\mu$ | ||||||||||||||||||||
| $\left(x-\mu\right)^2$ |
The sum of all $\left(x-\mu\right)^2 = \sum \left(x-\mu\right)^2 =$
The total number of data values, $n =$
Variance, $\sigma^2 = \frac{\sum \left(x-\mu\right)^2}{n} = $
Standard deviation, $\sigma = \sqrt{\frac{\sum \left(x-\mu\right)^2}{n}} = $
*Give your answer to 3 significant figures
*Give your answer to 3 significant figures
Example 3
Mean, $\mu$ =
Mean, $\mu$ =
| Data $x$ |
$3$ | $4$ | $4$ | $4$ | $4$ | $5$ | $7$ | $7$ | $7$ | $7$ | $7$ | $8$ | $8$ | $9$ | $9$ | $10$ | $10$ | $10$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $x-\mu$ | |||||||||||||||||||
| $\left(x-\mu\right)^2$ |
The sum of all $\left(x-\mu\right)^2 = \sum \left(x-\mu\right)^2 =$
The total number of data values, $n =$
Variance, $\sigma^2 = \frac{\sum \left(x-\mu\right)^2}{n} = $
*Give your answer to 3 significant figures
*Give your answer to 3 significant figures
Standard deviation, $\sigma = \sqrt{\frac{\sum \left(x-\mu\right)^2}{n}} = $
*Give your answer to 3 significant figures
*Give your answer to 3 significant figures
Summary
Range = $4$
IQR = $2$
Variance = $0.857$
Standard Deviation = $0.926$
Range = $6$
IQR = $0$
Variance = $1.71$
Standard Deviation = $1.31$
Range = $6$
IQR = $3$
Variance = $3.2$
Standard Deviation = $1.79$
Range = $7$
IQR = $5$
Variance = $5.37$
Standard Deviation = $2.32$
Range = $4$
IQR = $2$
Variance = $0.857$
Standard Deviation = $0.926$
Range = $6$
IQR = $0$
Variance = $1.71$
Standard Deviation = $1.31$
Range = $6$
IQR = $3$
Variance = $3.2$
Standard Deviation = $1.79$
Range = $7$
IQR = $5$
Variance = $5.37$
Standard Deviation = $2.32$
The greater the variance and standard deviation, the
spread the data is.