Summary of Comparing Variability of Datasets: 4B

This page would contain resource information like a glossary of terms from the section, key equations, and a reminder of concepts that were covered.

Make this more relevant to what students want — help them to build their processes, study guides, mnemonics, and memory dump material.

Essential Concepts

  • Variability can be measured in three ways: standard deviation, variance, and range.
  • Variability can be judged from a histogram by examining the distance of the bars from the statistical center (mean or median) of the graph. If the variability is high, equally sized or taller bars will appear away from the center of the graph. It the variability is low, the data will appear clustered around the center.
  • The following steps can be applied to calculate a standard deviation by hand:
    1. Calculate the mean of the population or sample.
    2. Take the difference between each data value and the mean. Then square each difference.
    3. Add up all the squared differences
    4. Divide by either the total number of observations in the case of a population or by 1 fewer than the total in the case of a sample.
    5. Take the square root of the result of the division in step 4.
  • Larger values of range indicate more variability in the data, but the range value only utilizes two observations in the entire dataset to measure variability. This is not an ideal measure of spread, but when used in combination with other measures of spread, it can help you gain a clearer understanding of the spread of a distribution.

Key Equations

  • Deviation from the mean

[latex]\left(x-\bar{x}\right)[/latex]
where [latex]\left(x\right)[/latex] is the observation in the dataset, and [latex]\left(\bar{x}\right)[/latex] is the sample mean.

  • Standard deviation of a population

[latex]\sigma = \sqrt{\dfrac{\sum \left(x-\mu\right)^2}{n}}[/latex]
where [latex]\sum[/latex] is the summation of [latex]{\left(x-\mu\right)^2}[/latex] for each observation, [latex]\left(x\right)[/latex] is the observation in the dataset, [latex]\left(\mu\right)[/latex] is the mean, and [latex]\left({n}\right)[/latex] is the number of observations.

  • Standard deviation of a sample

[latex]s=\sqrt{\dfrac{\sum \left(x-\bar{x}\right)^2}{n-1}}[/latex]
where [latex]\sum[/latex] is the summation of [latex]{\left(x-\bar{x}\right)^2}[/latex] for each observation, [latex]\left(x\right)[/latex] is the observation in the dataset, [latex]\left(\bar{x}\right)[/latex] is the mean, and [latex]\left({n}\right)[/latex] is the number of observations.

  • Variance of a population

[latex]\sigma^{2}=\dfrac{\sum\left(x-\mu\right)^{2}}{n}[/latex]
where [latex]\sum[/latex] is the summation of [latex]{\left(x-\mu\right)^2}[/latex] for each observation, [latex]\left(x\right)[/latex] is the observation in the dataset, [latex]\left(\mu\right)[/latex] is the mean, and [latex]\left({n}\right)[/latex] is the number of observations.

  • Variance of a sample

[latex]s^{2}=\dfrac{\sum\left(x-\bar{x}\right)^{2}}{n-1}[/latex]
where [latex]\sum[/latex] is the summation of [latex]{\left(x-\bar{x}\right)^2}[/latex] for each observation, [latex]\left(x\right)[/latex] is the observation in the dataset, [latex]\left(\bar{x}\right)[/latex] is the mean, and [latex]\left({n}\right)[/latex] is the number of observations.

Glossary

deviation from the mean
the distance between an observation ([latex]{x}[/latex]) in a dataset and the mean [latex]\left(\bar{x}\right)[/latex] of the dataset.
variability
a measure of how dispersed (spread out) the data are. It is often referred to as the spread, or dispersion, of a dataset.
standard deviation
a measure of how spread out observations are from the mean.
[latex]\sigma[/latex]
the standard deviation of a population of observations.
[latex]s[/latex]
the standard deviation of a sample of observations.
variance
the standard deviation squared.
[latex]\sigma^{2}[/latex]
the variance of a population of observations.
[latex]s^{2}[/latex]
the variation of a sample of observations.
range
the maximum (or largest) value – the minimum (or smallest) value.

Put formal DCMP I Can statements to prepare for the self-check.

These I Can Statements are new (the first one is the “you will understand” rephrased as an I Can):

  • I can describe variability using numerical summaries and its reflection in graphical displays.
  • I can use appropriate graphical displays and numerical summaries to describe variability of data with technology.
  • I can find and interpret the standard deviation of data.