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Wize AP Statistics Textbook > Calculating and Interpreting Statistics

Measure of Variability

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Range and Interquartile Range

Variability tells us how spread out the numbers are. They are useful for comparing two or more data sets containing quantitative variables to see if they have similar or different spread.

Range

Range=MAX−MIN\boxed{Range=MAX-MIN}
  • Weakest of all measures.
  • The range does not take into consideration how variable the data is in a given distribution.
  • Example:
  • "The exam grades range from 0% (min) to 100% (max)." That's all you know. You don't know how the grades are distributed between 0% and 100%.

Interquartile Range

Interquartile range (IQR)=Q3−Q1\boxed{Interquartile\,range\,(IQR)=Q_3-Q_1}
  • The interquartile range (IQR) is an improvement over “the range”.
  • Unlike the range, the IQR gives you an idea of how the data is distributed within quartiles.
  • 50% of the data falls within the interquartile range
  • 25% of the data is greater than Q3Q_3
  • 25% of the data is less than Q1Q_1
  • The IQR is a robust (resistant) measure, just like the median.










Wize Tip
The boxplot uses the interquartile range to display the distribution of quantitative data.


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Variance and Standard Deviation

The variance and standard deviation take into account all the data in the dataset and factor in how the data is distributed within it.

Sample Variance (s2)=∑(xi−x‾)2n−1\boxed{\text{Sample Variance}\,(s^{2 })=\frac{\sum_{ }^{ }\left(x_i-\overline{x}\right)^2}{n-1}}

Standard Deviation (s)=Variance\boxed{\text{Standard Deviation}\ \left(s\right)=\sqrt{\text{Variance}}}
  • σ2 \sigma^{2\ }is the population variance.
  • σ\sigma is the population standard deviation.
  • Standard deviation has the same units as the data.
  • Example
  • If the data are in centimetres, the standard deviation will be in centimetres.
  • If the data are in metres, the standard deviation will be in metres.
  • Variance has the same units as the data.

Which measure of variability is best?

  • For skewed data, the standard deviation and variance are easily influenced by outliers because they use all data, so the interquartile range, which is more resistant to extreme values, should be used.
  • For non-skewed/symmetric data, the standard deviation and variance are the best
Answer the following questions about variance and standard deviation.
Find the variance and standard deviation: 7 21 28 40

Calculate the range, interquartile range, and standard deviation:

12, 13, 13, 14, 19, 20, 34, 45, 66, 88, 90, 92, 200, 220

n=14n=14

x‾=66.1429\overline{x}=66.1429

∑(xi−x‾)2=60,435.7143\sum_{ }^{ }\left(x_i-\overline{x}\right)^2=60,435.7143



Practice: Standard Deviation

The data set is 20, 20, 30, 40, 40. Which of the following is/are true?

Check all that applies.
Extra Practice