2.2 Histograms
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This document discusses histograms and frequency distributions for organizing and summarizing data. It defines key terms like frequency distribution, class limits, and histograms. Histograms are described as bar graphs that visually display the shape, center, and spread of data distribution. Examples are provided to demonstrate how to construct histograms and interpret their features. Different common distribution shapes like normal, skewed left, and skewed right are illustrated. The document also introduces normal quantile plots as another tool to assess normality of data distribution by examining how closely the points follow a straight line pattern.
2.2 Histograms
- 1. Elementary Statistics Chapter 2: Exploring Data with Tables and Graphs 2.2 Histograms 1
- 2. Chapter 2: Exploring Data with Tables and Graphs 2.1 Frequency Distributions for Organizing and Summarizing Data 2.2 Histograms 2.3 Graphs that Enlighten and Graphs that Deceive 2.4 Scatterplots, Correlation, and Regression 2 Objectives: 1. Organize data using a frequency distribution. 2. Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives. 3. Represent data using bar graphs, Pareto charts, time series graphs, and pie graphs. 4. Draw and interpret a stem and leaf plot. 5. Draw and interpret a scatter plot for a set of paired data.
- 3. Recall : 2.1 Frequency Distributions for Organizing and Summarizing Data Data collected in original form is called raw data. Frequency Distribution (or Frequency Table) A frequency distribution is the organization of raw data in table form, using classes and frequencies. It Shows how data are partitioned among several categories (or classes) by listing the categories along with the number (frequency) of data values in each of them. Nominal- or ordinal-level data that can be placed in categories is organized in categorical frequency distributions. Lower class limits: The smallest numbers that can belong to each of the different classes Upper class limits: The largest numbers that can belong to each of the different classes Class boundaries: The numbers used to separate the classes, but without the gaps created by class limits Class midpoints: The values in the middle of the classes Each class midpoint can be found by adding the lower class limit to the upper class limit and dividing the sum by 2. Class width: The difference between two consecutive lower class limits in a frequency distribution Procedure for Constructing a Frequency Distribution 1. Select the number of classes, usually between 5 and 20. 2. Calculate the class width: 𝑊 = 𝑀𝑎𝑥−𝑀𝑖𝑛 # 𝑜𝑓 𝑐𝑙𝑎𝑠𝑠𝑒𝑠 and round up accordingly. 3. Choose the value for the first lower class limit by using either the minimum value or a convenient value below the minimum. 4. Using the first lower class limit and class width, list the other lower class limits. 5. List the lower class limits in a vertical column and then determine and enter the upper class limits. 6. Take each individual data value and put a tally mark in the appropriate class. Add the tally marks to get the frequency. 3
- 4. 2.2 Histograms While a frequency distribution is a useful tool for summarizing data and investigating the distribution of data, an even better tool is a histogram, which is a graph that is easier to interpret than a table of numbers. Key Concept Histogram: A graph consisting of bars of equal width drawn adjacent to each other (unless there are gaps in the data) The horizontal scale represents classes of quantitative data values, and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Important Uses of a Histogram • Visually displays the shape of the distribution of the data • Shows the location of the center of the data • Shows the spread of the data & Identifies outliers 4
- 5. 5 Example 1 Construct a histogram to represent the data for the record high temperatures for each of the 50 states. Class Limits Class Boundaries Frequency 100 - 104 105 - 109 110 - 114 115 - 119 120 - 124 125 - 129 130 - 134 99.5 - 104.5 104.5 - 109.5 109.5 - 114.5 114.5 - 119.5 119.5 - 124.5 124.5 - 129.5 129.5 - 134.5 2 8 18 13 7 1 1
- 6. Relative Frequency Histogram Relative Frequency Histogram has the same shape and horizontal scale as a histogram, but the vertical scale is marked with relative frequencies instead of actual frequencies. Time (seconds) Frequency 75-124 11 125-174 24 175-224 10 225-274 3 275-324 2 Example 2 Relative Frequency = f / n 11 / 50 = 0.22 24 / 50 = 0.48 10 / 50 = 0.2 3 / 50 = 0.06 2 / 50 = 0.04 Percent Frequency 22% 24% 20% 6% 4% 6 f = 50 rf = 1
- 7. Common Distribution Shapes 7 Critical Thinking Interpreting Histograms Explore the data by analyzing the histogram to see what can be learned about “CVDOT”: the Center of the data, the Variation, the shape of the Distribution, whether there are any Outliers, and Time.
- 8. Normal Distribution Because this histogram is roughly bell-shaped, we say that the data have a normal distribution. 8 Skewness: A distribution of data is skewed if it is not symmetric and extends more to one side than to the other. Data skewed to the right (positively skewed) have a longer right tail. Data skewed to the left (negative skewed) have a longer left tail.
- 9. 9 Example 3 Given Histogram: Find the following: 1. Estimate the number of subjects 2. The width 3. Boundaries of the 1st bar 4. Explain the gap. 5. Normal? Find the following: 1. The number of subjects 12 + 20 + 8 + 0 + 0 + 5 + 18 + 14 + 3 = 80 2. The width; 5.6 – 5.5 = 0.1 g 3. Boundaries of the 1st bar: 5.5g & 5.6g 4. The Gap: The 1st 40 quarters are from a different era than the 2nd set of 40 quarters. 5. Not Normally Distribution Pre 1964: 90% silver + 10 % copper Post 1964: Copper-Nickel alloy
- 10. Assessing Normality with Normal Quantile Plots Criteria for Assessing Normality with a Normal Quantile Plot Normal Distribution: The pattern of the points in the normal quantile plot is reasonably close to a straight line, and the points do not show some systematic pattern that is not a straight-line pattern. Not a Normal Distribution: The population distribution is not normal if the normal quantile plot has either or both of these two conditions: The points do not lie reasonably close to a straight-line pattern. The points show some systematic pattern that is not a straight-line pattern. 10 Normal Distribution: The points are reasonably close to a straight-line pattern, and there is no other systematic pattern that is not a straight-line pattern. Not a Normal Distribution: The points do not lie reasonably close to a straight line. Normal Quantile Plot: A normal quantile plot (or normal probability plot) is a graph of points (x, y) where each x value is from the original set of sample data, and each y value is the corresponding z score that is expected from the standard normal distribution.