Chapter 2 Notes: Exploring Data With Tables and Graphs
2-1 Frequency Distributions
Chapter focus: Exploring data with tables and graphs, including Frequency Distributions, Histograms, Graphs that Enlighten and Deceive, and Scatterplots/Correlation/Regression.
Key idea: When working with large data sets, it is often helpful to organize and summarize data by constructing a frequency distribution (a frequency table). Modern software and calculators can generate frequency distributions, so the details of constructing them are less important than what they tell us about the data.
Frequency Distribution (or Frequency Table) — definition:
Lists data values (either individually or by groups of intervals) with their corresponding frequencies (counts).
Purpose: summarize the data by showing how often each value or class occurs.
Data example used in tables:
Table 2-1: Academy Awards — Ages of Best Actresses and Best Actors (ages listed in order, from the first awards ceremony onward).
Frequencies for Best Actresses by age class (Table 2-2):
21-30: 28
31-40: 30
41-50: 12
51-60: 2
61-70: 2
71-80: 2
Total = 76
This data leads to a Frequency Distribution for the ages of Best Actresses.
Frequency distributions from Table 2-2 (Best Actresses):
Age group and frequency:
21-30 → 28
31-40 → 30
41-50 → 12
51-60 → 2
61-70 → 2
71-80 → 2
Total frequency:
Relative frequency distribution (same class limits as the frequency distribution, but using relative frequencies):
Relative frequency = class frequency ÷ sum of all frequencies.
Example (Table 2-3): Ages of Best Actresses (Relative Frequency Distribution):
21-30: 37% (i.e., rac{28}{76} = 0.368 ext{ or } 36.8 ext{%}
ightarrow 37 ext{%})31-40: 39%
41-50: 16%
51-60: 3%
61-70: 3%
71-80: 3%
Total relative frequencies sum to 100% (or 1.00).
Cumulative frequency distribution (Table 2-4):
Cumulative frequencies show the running total of frequencies up to the end of each class.
Example structure:
Less than 31: 28
Less than 41: 58
Less than 51: 70
Less than 61: 72
Less than 71: 74
Less than 81: 76
Interpretation: how many observations fall below the upper boundary of each class.
Constructing a frequency distribution (overview steps):
1) Decide on the number of classes (should be between 4 and 20).
2) Calculate class width (round up). A common guideline: where k is the number of classes.
3) Start with a lower limit for the first class.
4) Using the lower limit and class width, list the lower class limits.
5) List the lower class limits in a vertical column and enter the corresponding upper class limits.
6) Go through the data set, placing a tally in the appropriate class for each value.Why construct frequency distributions:
1) Large data sets can be summarized compactly.
2) They provide insight into the nature of the data.
3) They form the basis for constructing important graphs (histograms, ogives, etc.).Class concepts (definitions used when grouping data into classes):
Class Width: the difference between two consecutive lower class limits (or upper class limits, or class midpoints, or class boundaries). For example, with a typical grouping, the class width is often an equal interval like 10.
Lower Class Limits: the smallest values that can belong to a class.
Upper Class Limits: the largest values that can belong to a class.
Class Midpoints: the midpoints of each class, found by averaging the lower and upper class limits, . Example midpoints given: .
Class Boundaries: numbers that separate classes but without the gaps created by the class limits (often with .5 in decimals for integer data). Example boundaries: .
2-2 Histograms
Key concept: A histogram is an important type of graph that portrays the nature of the distribution.
Histogram definition:
A bar graph in which the horizontal scale represents the classes of data values and the vertical scale represents the frequencies.
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.
Normal distribution (bell-shaped):
One key characteristic is the bell shape; histograms can illustrate this.
Examples shown include normal (bell-shaped), uniform, and skewed distributions.
Interpreting histograms:
Use histograms to assess symmetry, skewness, modality, and the presence of outliers.
Example distributions shown:
Normal (bell-shaped) distribution (e.g., heights of 1000 women).
Uniform distribution (e.g., 1000 rolls of a die).
Skewed distribution (e.g., incomes of 1000 statistics professors).
On a regular histogram, you typically use class boundaries for the horizontal scale. The bars in a histogram are drawn such that their edges align with the class boundaries, allowing the bars to touch and represent the continuous nature of the data. Midpoints, however, are used for frequency polygons, where points are plotted above the midpoint of each class and then connected by line segments.
2-3 Graphs that Enlighten and Graphs that Deceive
Time-Series Graph:
Data collected at different points in time (x-axis: time; y-axis: the measured quantity).
Example: Number of Drive-in Movie Theaters over years (1987–2003).
Frequency Polygon:
Uses line segments connected to points directly above class midpoint values.
Based on the same frequency distribution as a histogram, but uses midpoints for the x-axis.
Example: Ages of Best Actresses with midpoints (e.g., 25.5, 35.5, 45.5, 55.5, 65.5, 75.5).
Ogive (Cumulative Frequency Plot):
A line graph that depicts cumulative frequencies.
X-axis typically shows class boundaries; Y-axis shows cumulative frequency.
Dot Plot:
Consists of a graph in which each data value is plotted as a point (or dot) along a scale of values.
Stemplot (Stem-and-Leaf Plot):
Represents data by separating each value into two parts: the stem (leftmost digits) and the leaf (rightmost digits).
Pareto Chart (Bar Graph):
Bar graph for qualitative data with bars arranged in order of decreasing frequency.
Pie Chart:
A graph depicting qualitative data as slices of a pie (proportions of categories).
Bad Graphs and Pictographs:
Examples showing how mis-leading graphics can distort interpretation (e.g., doubling length/width/height of a cube implies volume increases by eight; pictographs that exaggerate differences).
Examples shown in slides:
Pareto/bar charts for qualitative categories (e.g., access charges, international calling, operator services, marketing, cramming, slamming).
A typical salary chart with mis-leading scales labeled as "Bad Bar-Graphs".
2-4 Scatterplots, Correlation, and Regression
Scatterplot (Scatter Diagram):
A plot of paired (x, y) data with a horizontal x-axis and a vertical y-axis.
Properties of the Linear Correlation Coefficient r:
Range:
Invariance under scale changes: r does not change if all values of either variable are converted to a different scale.
Invariance under swapping x and y: If you interchange the roles of x and y, r remains the same.
r measures the strength of a linear relationship (the closer to ±1, the stronger the linear relationship).
Interpreting scatterplots with r values (examples):
Positive correlation examples: $r = 0.851$, $r = 0.991$, $r = 1$ (perfect positive).
Negative correlation examples: $r = -0.702$, $r = -0.965$, $r = -1$ (perfect negative).
Regression: The regression equation and line:
Regression equation is given by , which algebraically describes the relationship between the two variables.
The graph of the regression equation is called the regression line (also the line of best fit, or least squares line).
Interpretations from examples:
With sample data, the regression line and correlation coefficient quantify the strength and direction of the linear relationship.
Example: r = 0.926 (n = 8) indicates a significant positive linear correlation (there is enough evidence to reject the null hypothesis of no correlation in a typical test scenario, given the reported values).
Significance inference (example from slide):
There is a significant positive linear correlation (n = 8, r = 0.926).
In another example, r = -0.950 (n = 4) indicates a significant negative linear correlation.
Note: Specific p-values and critical values (CR/CV) may be shown on slides to illustrate the decision rule for significance, but the key takeaway is the sign and magnitude of r indicating the strength and direction of the linear relationship.
Quick recap of practical implications:
r summarizes linear association; it does not capture non-linear relationships.
Regression provides a predictive model of y from x, but extrapolation beyond the observed range should be done with caution.
Graphs (scatterplots, histograms, ogives, etc.) help diagnose data behavior and the appropriateness of linear models.
Important notes on the dataset and definitions encountered:
Time: Changing characteristics of the data over time (a recurring theme in time-series analysis).
Center (mean/average): A representative value indicating where the data are centered.
Variation: A measure of how much the values vary among themselves (spread, dispersion).
Distribution: The nature or shape of the data distribution (e.g., bell-shaped, uniform, skewed).
Outliers: Sample values that lie very far away from the vast majority of data values.
For large data sets, frequency distributions and related graphs provide essential summaries and visual insights that complement numerical measures.
Notation summary:
Class width: , where k is the number of classes.
Class midpoints: .
Class boundaries: defined to separate classes without gaps (e.g., ).
Relative frequency: .
Cumulative frequency: running total of frequencies.
Regression line: .
Correlation coefficient: .
Quick references to data formats seen in the slides:
Table 2-1 and Table 2-2 illustrate the Ages of Best Actresses and Best Actors; Table 2-3 shows the Relative Frequency Distribution; Table 2-4 shows the Cumulative Frequency Distribution.
Histograms and related graphs use class intervals (e.g., 21-30, 31-40, etc.) with frequencies or relative frequencies.
Dot plots, stem-and-leaf plots, Pareto charts, and pie charts offer alternative visualization methods for different data types.
While a histogram uses the classes of data values on its horizontal scale, the concepts of midpoints and class boundaries are distinctly used in other related graphs:
Frequency Polygons use line segments connected to points directly above class midpoint values.
Ogives (Cumulative Frequency Plots) typically show class boundaries on their x-axis.
Class boundaries are also fundamental to the definition of the classes themselves in a frequency distribution, as they define the numbers that separate classes without gaps (e.g., ).
On a regular histogram, you typically use class boundaries for the horizontal scale. The bars in a histogram are drawn such that their edges align with the class boundaries, allowing the bars to touch and represent the continuous nature of the data. Midpoints, however, are used for frequency polygons, where points are plotted above the midpoint of each class and then connected by line segments.