Histograms

A histogram is a graphical method for displaying the shape of a distribution.
Particularly useful for large datasets, such as when there are many observations.

Example of creating a histogram is provided using scores of 642 students on a psychology test.
The test comprises 197 items, graded as either correct or incorrect.
The range of student scores is from 46 to 167.

The first step is to create a frequency table.
A simple frequency table for all scores would be too large, exceeding 100 rows.
To simplify, scores need to be grouped into class intervals.
The intervals are:
- First interval: 39.5 to 49.5
- Second interval: 49.5 to 59.5
- Subsequent intervals follow a similar pattern.
Class Intervals: Ranges into which data points are grouped to create a manageable dataset.
Class intervals help condense the data into comprehensible sections.

Scores falling within each interval were counted to obtain class frequencies.
Example counts:
- First interval (39.5-49.5): 3 scores
- Second interval (49.5-59.5): 10 scores
- Additional counts would complete the table.
- The frequency table has a left column for class intervals and a right column for frequency counts.

To create a histogram based on the frequency table:
- The count in each interval is represented as a bar of height corresponding to the frequency.
The histogram displays the first interval (39.5-49.5) with three scores and another key interval (79.5-89.5) with the highest counts of 147 scores.
Each bar's height reflects the number of scores in its corresponding interval.

The resulting histogram shows:
- A predominance of scores in the middle of the distribution.
- A lower frequency of scores at the extremes.
The distribution is noted to be asymmetrical with a positive skew:
- Scores extend further to the right than to the left, indicating that there are more higher scores.

Histograms can also manage continuous data (e.g., time measured to fractions of seconds).
For instance, a task may require measuring time down to the thousandths of a second.
When dealing with continuous data:
- There's generally no need to consider values exactly at class boundaries since precision allows for more flexibility.
Example of class boundaries with whole numbers: 4,000, 5,000 milliseconds.
- Each class frequency measures the number of observations within specified bounds, such as 4,000 – 4,999 milliseconds.

A variation in histograms can use relative frequencies:
- This represents the proportion of scores per interval rather than mere counts.
- The y-axis in such a histogram ranges from 0 to 1.
To convert a frequency histogram to a relative frequency histogram:
- Step A: Divide each class frequency by the total number of observations.
- Step B: Plot the resulting quotients on the y-axis labeled as proportions.

Calculate the range of scores: Subtract the smallest observation from the largest observation.
Break the range into class intervals.
Count the number of scores in each interval to obtain class frequencies.
Represent these frequencies using bars in a histogram.

A further example examines response times to a task over 20 trials measured in milliseconds.
Each time is sorted from lowest to highest.
The variable being measured, time to respond, is characterized as a continuous variable:
- No two response times are expected to be identical due to the precision of measurement.
An ungrouped frequency distribution of the response times would be unhelpful:
- It would consist of 20 distinct times without revealing any frequency relationship, each time having a count of one.
Understanding relationships can be achieved by constructing a corresponding histogram.
A table prepared with tally marks helps visualize the frequency for each duration.
The class interval with the highest frequency (e.g., 600 to 700 milliseconds) corresponds to the tallest bar in the histogram, indicating the most frequent response time.