1.3 Frequency, Frequency Tables, and Levels of Measurement

Frequency and Frequency Tables

Frequency tables organize raw data into a structured format so you can see patterns at a glance. Instead of staring at a long list of numbers or categories, you get a clear summary of how often each value appears, what proportion of the data it represents, and how observations accumulate across categories. This section covers how to build and interpret these tables, and how the type of data you're working with affects what you can do with it.

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Relative and Cumulative Frequencies

Frequency is simply the count of how many times a particular value or category appears in your dataset. Two extensions of basic frequency come up constantly in statistics:

• Relative frequency is the proportion of observations in a given category. You calculate it by dividing the category's frequency by the total number of observations. This lets you compare across datasets of different sizes. For example, if 40 out of 200 students are sophomores, the relative frequency is $\frac{40}{200} = 0.20$, or 20%.
• Cumulative frequency is the running total of frequencies as you move through the categories in order. You add each category's frequency to the sum of all previous categories. This tells you how many observations fall at or below a certain point. If 30 freshmen, 40 sophomores, and 25 juniors are in a dataset, the cumulative frequency through juniors is $30 + 40 + 25 = 95$.

You can also compute cumulative relative frequency by dividing the cumulative frequency by the total number of observations. All relative frequencies in a table should sum to 1.0 (or 100%), and the final cumulative relative frequency should also equal 1.0.

Levels of Measurement

The level of measurement for your data determines which statistical operations make sense. There are four levels, and each one builds on the previous:

• Nominal: Categories or labels with no inherent order. You can count frequencies and find the mode, but you can't rank or average them. Examples: eye color, zip code, type of pet.
• Ordinal: Categories that have a natural ranking, but the distances between ranks aren't necessarily equal. You know that "A" is better than "B," but you can't say the gap between A and B is the same as between B and C. Examples: letter grades, satisfaction ratings (poor/fair/good/excellent), class rank.
• Interval: Numerical data where the differences between values are consistent and meaningful, but there's no true zero point. A temperature of 0°F doesn't mean "no temperature," so you can't say 80°F is "twice as hot" as 40°F. Examples: temperature in Celsius or Fahrenheit, calendar year, SAT scores.
• Ratio: Numerical data with both meaningful differences and a true zero point, which makes ratios valid. A person who weighs 200 lbs genuinely weighs twice as much as someone who weighs 100 lbs. Examples: height, weight, income, age, distance.

Grouped Frequency Tables

When your data is continuous (meaning it can take on any value within a range, like height or time), listing every individual value would make a massive, unhelpful table. Instead, you group the data into classes (intervals) and count how many observations fall into each one.

Constructing a Grouped Frequency Table

Follow these steps:

1. Find the range: Subtract the smallest value from the largest. If your data runs from 12.3 to 87.9, the range is $87.9 - 12.3 = 75.6$.

2. Choose the number of classes: Typically between 5 and 20. Too few classes hides patterns; too many defeats the purpose of grouping.

3. Calculate the class width: Divide the range by the number of classes and round up to a convenient number. For 8 classes: $\frac{75.6}{8} = 9.45$, so you might round up to 10.

4. Set the class boundaries: Start at or just below the minimum value and add the class width repeatedly. Make sure every data point falls into exactly one class with no gaps or overlaps.

5. Tally the frequencies: Count how many observations fall into each class.

Components of a Grouped Frequency Table

Each grouped frequency table typically includes:

• Class boundaries: The lower and upper limits defining each interval
• Class midpoints: The average of the lower and upper boundaries for each class, calculated as $\frac{\text{lower} + \text{upper}}{2}$. These are useful when you need a single value to represent the class (for example, when computing a weighted mean).
• Frequency: The count of observations in each class
• Relative frequency: The proportion of total observations in each class
• Cumulative frequency: The running total of frequencies through each class

Once built, a grouped frequency table gives you a clear picture of how your data is distributed and serves as the basis for visual displays like histograms.