Descriptive Statistics

Chapter Outline

• 2.1 Frequency Distributions and Their Graphs

• 2.2 More Graphs and Displays

• 2.3 Measures of Central Tendency

• 2.4 Measures of Variation

• 2.5 Measures of Position

Frequency Distributions and Their Graphs

Frequency Distribution

A frequency distribution is a table that organizes data into classes or intervals, showing the number of data entries (frequency, f) in each class. This helps to summarize large data sets and identify patterns.

• Lower class limit: The smallest value that can belong to a class.

• Upper class limit: The largest value that can belong to a class.

• Class width: The difference between the lower (or upper) limits of consecutive classes.

• Range: The difference between the maximum and minimum data entries.

Constructing a Frequency Distribution

1. Decide on the number of classes (usually 5–20).

2. Find the class width: Round up to the next convenient number.

3. Find the class limits:

   • Start with the minimum value as the first lower class limit.

   • Add the class width to get subsequent lower limits.

   • Upper class limit of a class is one less than the lower limit of the next class.

4. Tally each data entry in the appropriate class and count the frequencies.

Example: For cell phone screen times (in minutes) for 30 adults, with a minimum of 155 and maximum of 405, and 7 classes, the class width is rounded up to 36.

Midpoints, Relative Frequency, and Cumulative Frequency

• Midpoint: The value halfway between the lower and upper limits of a class.

• Relative frequency: The proportion or percentage of data in a class.

• Cumulative frequency: The sum of the frequencies for that class and all previous classes. The cumulative frequency of the last class equals the sample size.

Example: In the cell phone screen time data, the most common range is 299 to 334 minutes, and about half the times are less than 299 minutes.

Frequency Histogram

A frequency histogram is a bar graph representing the frequency distribution. The horizontal axis is quantitative (data values), and the vertical axis shows frequencies. Bars are adjacent (touching), reflecting the continuous nature of the data.

• Class boundaries: Values that separate classes without gaps, ensuring bars touch. Calculated as the average of the upper limit of one class and the lower limit of the next.

Example: A histogram for cell phone screen times shows that two-thirds of adults spend more than 262.5 minutes per day on their phones.

Frequency Polygon

A frequency polygon is a line graph that emphasizes the continuous change in frequencies. It is constructed by plotting points at the class midpoints (horizontal axis) and frequencies (vertical axis), then connecting them with straight lines. The graph starts and ends at the horizontal axis, one class width before the first and after the last midpoint.

Example: The frequency polygon for cell phone screen times shows frequency increases up to about 316.5 minutes, then decreases.

Relative Frequency Histogram

A relative frequency histogram has the same shape and horizontal scale as the frequency histogram, but the vertical axis measures relative frequencies (proportions or percentages) instead of raw frequencies.

Example: 20% of adults have screen times between 262.5 and 298.5 minutes, which may not be obvious from the frequency histogram alone.

Cumulative Frequency Graph (Ogive)

An ogive is a line graph displaying cumulative frequencies at each upper class boundary. The horizontal axis marks upper class boundaries, and the vertical axis shows cumulative frequencies. The graph starts at the lower boundary of the first class (cumulative frequency zero) and ends at the upper boundary of the last class (cumulative frequency equals sample size).

Example: The ogive for cell phone screen times shows that 10 adults had screen times of 262.5 minutes or less, with the steepest increase between 298.5 and 334.5 minutes.

Using Technology to Construct Graphs

Statistical software and calculators (e.g., TI-84 Plus, Minitab, Excel, StatCrunch) can be used to construct histograms and other graphs. Data (such as midpoints and frequencies) are entered into lists, and the software generates the appropriate graph.

• For the TI-84 Plus: Enter midpoints in L1, frequencies in L2, and use the histogram plot feature.

• Other software (Minitab, Excel, StatCrunch) provides similar functionality for creating frequency histograms.