Frequency Distributions and Histograms in Statistics

Frequency Distributions

Introduction to Frequency Distributions

A frequency distribution is a table that organizes data values into classes or intervals, showing the number of data entries (frequency) in each class. This is a foundational concept in descriptive statistics, helping to summarize large data sets and reveal patterns.

• Frequency (f): The number of data entries in a class.

• Class: A group or interval into which data values are sorted.

• 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 distance between consecutive lower (or upper) class limits.

Formula for Class Width:

$\text{Class Width} = \frac{\text{Range}}{\text{Number of Classes}}$

Always round up to the next whole number when calculating class width.

Constructing a Frequency Distribution

1. Determine the number of classes (often between 5 and 20 for most data sets).

2. Find the class limits (lower and upper) for each class.

3. Calculate the class width.

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

Example Frequency Distribution Table:

Class	Frequency, f
1–5	5
6–10	8
11–15	8
16–20	5
21–25	4

• Lower class limits: 1, 6, 11, 16, 21, 26

• Upper class limits: 5, 10, 15, 20, 25, 30

• Class width: $6 - 1 = 5$

Detailed Example: Constructing a Frequency Distribution

Suppose we have data on the response times (in minutes) for U.S. adults. To construct a frequency distribution with seven classes:

• Range: $405 - 155 = 250$

• Class width: $\frac{250}{7} \approx 35.71$, round up to 36

Class	Tally	Frequency	Midpoint	Relative Frequency	Cumulative Frequency
155–190	III	3	172.5	0.1	3
191–226	II	2	208.5	0.07	5
227–262	III	3	244.5	0.1	8
263–298	IIII	4	280.5	0.13	12
299–334	IIII	4	316.5	0.13	16
335–370	IIIIIII	7	352.5	0.23	23
371–406	IIIIIII	7	388.5	0.23	30

• Midpoint: $\frac{\text{LCL} + \text{UCL}}{2}$ (LCL = lower class limit, UCL = upper class limit)

• Relative Frequency: $\frac{\text{class frequency}}{\text{sample size}}$

• Cumulative Frequency: The sum of the frequencies for that class and all previous classes.

Key Properties:

• The distance between midpoints equals the class width.

• The sum of all frequencies equals the sample size ($n$).

Histograms

Introduction to Histograms

A histogram is a graphical representation of a frequency distribution. It uses adjacent bars to show the frequency of data within each class interval. Histograms are useful for visualizing the shape, center, and spread of data.

• The horizontal axis represents the quantitative data (class intervals).

• The vertical axis represents the frequencies of the classes.

• Bars must be adjacent (touching) to indicate continuous data.

Example Histogram

Age Interval	Frequency
20–29	2
30–39	6
40–49	7
50–59	5

The corresponding histogram would have bars for each age interval, with heights proportional to the frequencies (2, 6, 7, 5).

Constructing a Frequency Histogram (StatCrunch Example)

1. Select "Graph" in StatCrunch.

2. Select "Histogram".

3. Choose the data you want to plot in the histogram.

Summary Table: Key Terms and Formulas

Term	Definition	Formula
Class Width	Distance between consecutive lower (or upper) class limits	$\frac{\text{Range}}{\text{Number of Classes}}$
Midpoint	Center value of a class	$\frac{\text{LCL} + \text{UCL}}{2}$
Relative Frequency	Proportion of data in a class	$\frac{f}{n}$
Cumulative Frequency	Sum of frequencies up to a class	Sum of frequencies for that class and all previous classes

Example Application

Suppose you have a data set of exam scores for 30 students. To summarize the data, you could:

• Organize the scores into 5–7 classes using the class width formula.

• Construct a frequency distribution table.

• Calculate midpoints, relative frequencies, and cumulative frequencies.

• Draw a histogram to visualize the distribution of scores.

Additional info: These methods are foundational for further statistical analysis, such as calculating measures of central tendency and dispersion.