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Distribution Analysis: Modes and Symmetry

This analysis examines the distribution of data to determine the presence of modes and symmetry. It includes a histogram, ogive, and percentile calculations.

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Distribution Analysis: Modes and Symmetry

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  1. Does the distribution have one or more peaks (modes) or is it unimodal? • Is the distribution approximately symmetric or is it skewed in one direction? Is it skewed to the right (right tail longer) or left?

  2. Example Description • Shape: The distribution is roughly symmetric with a single peak in the center. • Center: You can see from the histogram that the midpoint is not far from 110. The actual data shows that the midpoint is 114. • Spread: The spread is from 80 to about 150. There are no outliers or other strong deviations from the symmetric, unimodal pattern.

  3. Calculator Example Text (To save data for later use on home screen type L1 -> Prez)

  4. Calc continued • Frequency shortcut: If you have a dataset comprised of 75 3’s and 35 4’s for example, you can enter the values in list 1 and the frequencies in list 2 then pull 1 variable stats: • Stats-edit- L1: 3, 4 L1: 75, 35 stat-calc-1var stats L1,L2 enter

  5. Relative frequency/Cumulative Frequency • A histogram does a good job of displaying the distribution of values of a quantitative variable, but tells us little about the relative standing of an individual observation. • So, we construct an ogive (“Oh-Jive”) aka a relative cumulative frequency graph.

  6. Step 1- Construct table • Decide on intervals and make a frequency table with 4 columns: Freq, Relative frequency, cumulative frequency, and rel. cum. Freq. • To get the values in the rel. freq. column, divide the count in each class interval by the total number of observations. Multiply by 100 to convert to %. • In Cum freq column, add the counts that fall in or below the current class interval • for rel. cum. freq. column, divide the entries in the cum freq column by total number of individuals.

  7. Step 2 & 3 • Label and scale your axes and title your graph. Vertical axis always Relative Cum. Freq. Scale the horizontal axis according to your choice of class intervals and the vertical axis from 0% to 100%. • Plot a point corresponding to the rel. Cum. freq. in each class interval at the LEFT ENDPOINT of the NEXT class interval. (example, the 40 to 44 interval, plot a point at a height of 4.7% above the age value of 45. • Begin with 0% you should end with 100%. Connect dots

  8. To Locate an individual within distribution: What about Clinton? He was 46. To find his relative standing, draw a vertical line up from his age (46) on the horizontal axis until it meets the ogive. Then draw a horizontal line from this point of intersection to the vertical axis. Based on our graph his age places him at the 10% mark which tells us that about 10% of all US presidents were the same age as or younger than Bill Clinton when they were inaugurated. To locate a value corresponding to a percentile, do the opposite. Ex: 50th percentile, 55 years old.

  9. Whenever data are collected over time, plot observations in time order. Displays of distributions such as stemplots and histograms which ignore time order can be misleading when there is systematic change over time.

  10. Shows change in gas price over time. Shows TRENDS

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