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Describing Distributions with Numbers

Learn about measures of center and spread in descriptive statistics, including mean, median, mode, quartiles, standard deviation, and variance.

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Describing Distributions with Numbers

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  1. Lesson 1 - 2 Describing Distributions with Numbers parts from Mr. Molesky’s Statmonkey website

  2. Knowledge Objectives • What is meant by a resistant measure? • Two reasons why we use squared deviations rather just average deviations from the mean • What is meant by degrees of freedom”

  3. Construction Objectives • Identify situations in which the mean is the most appropriate measure of center and situations in which the median is the most appropriate measure • Given a data set: • Find the quartiles • Find the five-number summary • Compute the mean and median as measures of center • Compute the interquartile range (IQR) • Use the 1.5IQR rule to identify outliers • Compute the standard deviation and variance as measures of spread

  4. Construction Objectives cont • Identify situations in which the standard deviation is the most appropriate measure of spread and situations in which the interquartile range is the most appropriate measure • Explain the effect of a linear transformation of a data set on the mean, median, and standard deviation of the set • Use numerical and graphical techniques to compare two or more data sets

  5. Vocabulary • Mean – the average value • Median – the middle value (in an ordered list) • Resistant measure – a measure (statistic or parameter) that is not sensitive to the influence of extreme observations • Mode – the most frequent data value • Range – difference between the largest and smallest observations • Pth percentile – p percent of the observations(in an ordered list) fall below at or below this number • Quartile – multiples of 25th percentile (Q1 – 25th ; Q2 –50th or median; Q3 – 75th) • Five number summary – the minimum, Q1, Median, Q3, maximum

  6. Vocabulary cont • Boxplot – graphs the five number summary and any outliers • Interquartile range (IQR) – where IQR = Q3 – Q1 • Outlier – a data value that lies outside the interval [Q1 – 1.5IQR, Q3 + 1.5IQR] • Variance – the average of the squares of the deviations from the mean • Standard Deviation – the square toot of the variance • Degrees of freedom – the number of independent pieces of information that are included in your measurement • Linear transformation – changes the data in the form of xnew = a + bx

  7. Measures of Center • Numerical descriptions of distributions begin with a measure of its “center” • If you could summarize the data with one number, what would it be? Mean: The “average” value of a dataset Median: The “middle” value of an ordered dataset Arrange observations in order min to max Locate the middle observation, average if needed.

  8. Mean vs Median • The mean and the median are the most common measures of center • If a distribution is perfectly symmetric, the mean and the median are the same • The meanis not resistant to outliers • The mode, the data value that occurs the most often, is a common measure of center for categorical data • You must decide which number is the most appropriate description of the center... Mean Median Applet • http://bcs.whfreeman.com/tps3e/content/cat_020/applets/meanmedian.html • Use the mean on symmetric data andthe median on skewed data or data with outliers

  9. Distributions Parameters Median Mean Mode Mean < Median < Mode Skewed Left: (tail to the left) Mean substantially smaller than median(tail pulls mean toward it)

  10. Distributions Parameters Mode Median Mean Mean ≈ Median ≈ Mode Symmetric: Mean roughly equal to median

  11. Distributions Parameters Median Mode Mean Mean > Median > Mode Skewed Right: (tail to the right) Mean substantially greater than median(tail pulls mean toward it)

  12. Central Measures Comparisons

  13. Example 1 Which of the following measures of central tendency resistant? • Mean • Median • Mode Not resistant Resistant Resistant

  14. Example 2 Given the following set of data:70, 56, 48, 48, 53, 52, 66, 48, 36, 49, 28, 35, 58, 62, 45, 60, 38, 73, 45, 51,56, 51, 46, 39, 56, 32, 44, 60, 51, 44, 63, 50, 46, 69, 53, 70, 33, 54, 55, 52What is the mean?What is the median?What is the mode? What is the shape of the distribution? 51.125 51 48, 51, 56 Symmetric(tri-modal)

  15. Example 3 Given the following types of data and sample sizes, list the measure of central tendency you would use and explain why? Sample of 50 Sample of 200Hair colorHeightWeightParent’s IncomeNumber of SiblingsAgeDoes sample size affect your decision? mode mode mean mean mean mean median median mean mean mean mean Not in this case, but the larger the sample size, might allow use to use the mean vs the median

  16. Shape? Outliers? Center? Spread? Sample Data Consider the following test scores for a small class: Plot the data and describe the SOCS: What number best describes the “center”? What number best describes the “spread’?

  17. Day 1 Summary and Homework • Summary • Three characteristics must be used to describe distributions (from histograms or similar charts) • Shape (uniform, symmetric, bi-modal, etc) • Center (mean, median, mode measures) • Spread (variance – next lesson) • Median is resistant to outliers; mean is not! • Use Mean for symmetric data • Use Median for skewed data (or data with outliers) • Use Mode for categorical data • Homework • pg 74 – 75: problems 27-31

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