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Understanding Histograms and Ogives for Data Analysis

Learn how to create frequency tables, histograms, and Ogive graphs to visualize and analyze quantitative data effectively. Explore various distribution shapes and types of histograms for data interpretation.

timothyfranklin
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Understanding Histograms and Ogives for Data Analysis

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  1. Section • 2.1 • Frequency Distributions, Histograms, and Related Topics

  2. Frequency Tables • For quantitative data, these tables • organize data into smaller intervals or classes • display how many data values fall into each class

  3. Steps • 1. First decide how many classes you want (5 to 15 usually). • 2. Next, find the class width for the classes and round up.

  4. Get the class limits: • a) use smallest data value as the lower class limit of the 1stclass. • b) Add the class width to it to find that the lower class limit for the 2ndclass. • c) Following this pattern until you have all lower limits • d) fill in the upper class limits to span the data

  5. Record the # of data points within each class in a frequency, f, column • Calculate relative frequency for each class in a separate column (total approximately  1) • 6. Sometimes you’re asked for the class midpoints, often used as a representative value of the entire class.

  6. 7. There’s a space between classes. To fill them, find the class boundaries because they’re used for histograms • 8. Find the cumulative frequency for an OGIVE graph

  7. Example 1 – Make a Frequency table • A task force to encourage car pooling did a study of one-way commuting distances of workers in the downtown Dallas area. A random sample of 60 of these workers was taken. The commuting distances of the workers in the sample are given in Table. • One-Way Commuting Distances (in Miles) for 60 Workers in Downtown Dallas

  8. Example 1

  9. Histograms • provide a great display for the shape of the data • use bars to represent each class • width of each bar is the class width • the markers on the x-axis are the class boundaries • the height of the bar (y-axis) can be the class frequency or the relative frequency (percent) of that class

  10. Example 2 – Make a Histogram • Using data from example 1:

  11. Histograms for example 1: • Histogram for Dallas Commuters: • One-Way Commuting Distances • Relative-Frequency Histogram for Dallas Commuters: One-Way Commuting Distances

  12. Distribution Shapes • Distribution: collection of numbers • If the raw data came from a random sample, the histogram should have a similar shape to that of the population. Bell-shaped: The highest frequency class (tallest bar) is in the middle while other classes decrease symmetrically around it (one hill) Uniform: every class has equal frequency (bars of roughly equal height) (no hill)

  13. Distribution Shapes • Right-Skewed: the longer tail of the histogram trails to the right. Mound on the left. • Left-Skewed: the longer tail of the histogram trails to the left. Mound on the right. • Bimodal: two classes have the largest frequencies while separated by other lower frequency classes (two hills)

  14. Distribution Shapes • Types of Histograms

  15. Distribution Shapes • Types of Histograms

  16. Purpose of Histograms • At one glances if the distribution is… • symmetric, skewed, or bimodal? • has outliers? (values different from other data values) • And… • which classes contain the most data • how spread out the data are

  17. Ogives • (“oh-jive”)is a graph that displays cumulative frequencies (y-axis) • Or cumulative percent (y-axis) if you divide cumulative frequencies by total # of data • The markers on the x-axis are the class boundaries • Start on the x-axis and put a dot over each boundary to indicate the height of the cumulative frequency • Connect dots with a line segments

  18. Purpose of Ogives • how many data (or percent) are below a value on the x-axis • how the data values accumulate over the range of the data

  19. Example 3 – Make an Ogive Graph • Aspen, Colorado, is a world-famous ski area. If the daily high temperature is above 40F, the surface of the snow tends to melt. It then freezes again at night. This can result in a snow crust that is icy. It also can increase avalanche danger. Table gives a summary of daily high temperatures (F) in Aspen during the 151-day ski season.

  20. Example 3 – Ogive • Ogive for Daily High Temperatures (F) During Aspen Ski Season

  21. Example 3 • Estimate the total number of days with a high temperature lower than or equal to 40F: • Solution: • 117 days have had high temperatures of no more than 40F.

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