Describing Data with Tables, Graphs, and Numerical Measures

Describing Data with Tables and Graphs

Class Intervals, Frequency, and Relative Frequency

Organizing raw data into frequency tables is a foundational step in descriptive statistics. This process helps summarize large datasets and reveals patterns in the data.

• Class Limits: The lower and upper boundaries for each interval in a frequency table.

• Frequency: The number of data values falling within each class interval.

• Relative Frequency: The proportion of data values in each class interval, calculated as: $\text{Relative Frequency} = \frac{\text{Class Frequency}}{\text{Total Frequency}}$

• Example Table:

Class Limits	Frequency	Relative Frequency
1-8	14	0.25
9-16	21	0.38
17-24	11	0.20
25-32	4	0.07
33-40	4	0.07
41-48	1	0.02

Additional info: Relative frequencies are rounded to two decimal places for clarity.

Class Midpoints and Boundaries

Class midpoints and boundaries are used to further analyze grouped data and to construct histograms.

• Class Midpoint: Calculated as the average of the lower and upper class limits. $\text{Class Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}$

• Class Boundaries: To find upper boundaries, add 0.5 to the upper class limit; to find lower boundaries, subtract 0.5 from the lower class limit.

• Example Table:

Class Limits	Class Boundaries	Frequency	Class Midpoint
1-8	0.5-8.5	14	4.5
9-16	8.5-16.5	21	12.5
17-24	16.5-24.5	11	20.5
25-32	24.5-32.5	4	28.5
33-40	32.5-40.5	4	36.5
41-48	40.5-48.5	1	44.5

Histograms and Frequency Polygons

Histograms and frequency polygons are graphical representations of frequency distributions.

• Histogram: A bar graph where the height of each bar represents the frequency or relative frequency of each class interval.

• Frequency Polygon: A line graph that connects the midpoints of each class interval at heights corresponding to their frequencies.

• Steps to Create:

  1. Draw horizontal axis with class boundaries or midpoints.

  2. Draw vertical axis with frequencies or relative frequencies.

  3. For histograms, draw bars for each class; for polygons, connect points at each midpoint.

Cumulative Frequency

Cumulative frequency is the running total of frequencies through the classes in a frequency distribution.

• Calculation: Add each frequency to the sum of the previous frequencies.

• Example Table:

Frequency	Cumulative Frequency
43	43
23	66

Describing Data Numerically

Stem-and-Leaf Displays

Stem-and-leaf plots are a method for displaying quantitative data in a way that retains the original data values and shows their distribution.

• Stem: The leading digit(s) of each data value.

• Leaf: The trailing digit(s) of each data value.

• Steps:

  1. Split each number into stem and leaf.

  2. List stems in a column, leaves to the right.

  3. Order leaves from smallest to largest.

  4. Label the plot to indicate the value of the stems and leaves.

• Example: Displaying weights of carry-on luggage in pounds.

Measures of Central Tendency

Central tendency describes the center of a data set using mean, median, and mode.

• Mean (Arithmetic Average): The sum of all data values divided by the number of values. $\bar{x} = \frac{\sum x_i}{n}$

• Median: The middle value when data is ordered. If even number of values, median is the average of the two middle values.

• Mode: The value that occurs most frequently in the data set. A set may have no mode, one mode, or multiple modes.

• Example: For the data set [20, 22, 22, 23, 24], the mean is $\bar{x} = \frac{20+22+22+23+24}{5} = 22.2$, the median is 22, and the mode is 22.

Position of the Median in Large Data Sets

To find the position of the median in a large data set:

• Use the formula: $\text{Position of the median} = \frac{n+1}{2}$, where $n$ is the total number of data values.

• Example: For $n = 20$, position is $\frac{20+1}{2} = 10.5$ (median is average of 10th and 11th values).

Trimmed Mean

A trimmed mean is calculated by removing a specified percentage of the smallest and largest values before computing the mean.

• Steps:

  1. Order data from smallest to largest.

  2. Remove the lowest and highest values according to the desired percentage.

  3. Calculate the mean of the remaining data.

• Example: For a 5% trimmed mean in a sample of 20, remove the lowest and highest value, then compute the mean of the remaining 18 values.

Additional info:

• These notes cover foundational techniques for organizing and summarizing data, including frequency tables, histograms, stem-and-leaf plots, and measures of central tendency, which are essential for introductory statistics courses.