Frequency Tables and Histograms

When you collect a large set of numbers, looking at every individual value is overwhelming. A frequency table organizes raw data by counting how often each value (or range of values) occurs. A histogram is the visual version — a bar chart specifically designed for showing frequency distributions of continuous data.

Building a Frequency Table from Raw Data

Start with raw data and tally how many times each value appears.

Example 1: Quiz Scores

A class of 20 students received these quiz scores (out of 10):

$7, 8, 9, 7, 6, 8, 10, 7, 8, 9, 5, 7, 8, 6, 9, 8, 7, 10, 8, 9$

Step 1: List every unique value and count its occurrences.

Step 2: Verify the frequencies sum to the total number of data points.

$1 + 2 + 5 + 6 + 4 + 2 = 20 \checkmark$

This table immediately shows that 8 was the most common score (the mode) and that most students scored 7 or above.

Class Intervals (Bins)

When data has many distinct values or is continuous (like heights, weights, or temperatures), grouping values into class intervals (also called bins) makes the table manageable.

Guidelines for choosing intervals:

• Use 5 to 10 intervals for most datasets
• Make all intervals the same width
• Intervals should not overlap — use ranges like 60–69, 70–79 (or 60 to under 70, 70 to under 80)
• Every data point must fall into exactly one interval

Example 2: Employee Ages

A company has 25 employees with these ages:

$22, 25, 28, 31, 33, 35, 36, 38, 40, 41, 42, 43, 44, 45, 47, 48, 50, 52, 53, 55, 56, 58, 60, 62, 64$

Using intervals of width 10:

Relative and Cumulative Frequency

Relative frequency expresses each frequency as a fraction or percentage of the total. It answers: “What proportion of the data falls in this interval?”

$\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Number of Values}} \times 100$

Cumulative frequency is a running total — it answers: “How many data points fall at or below this interval?”

Using the employee age data:

From the cumulative column, we can quickly see that 16 out of 25 employees (64%) are 49 or younger, and 22 out of 25 (88%) are 59 or younger.

Histograms vs. Bar Charts

Histograms and bar charts look similar but serve different purposes:

In a histogram, bars touch because the data is continuous — the end of one interval is the start of the next. The height of each bar represents the frequency (or relative frequency) of that interval. The shape of the histogram reveals whether data is symmetric, skewed left, skewed right, or uniform.

Calculating the Mean from a Frequency Table

You can estimate the mean from a frequency table without going back to the raw data.

$\text{Mean} = \frac{\sum (value \times frequency)}{\sum frequency}$

For grouped data, use the midpoint of each interval as the value.

Using the employee age data:

$\text{Estimated Mean} = \frac{1122.5}{25} = 44.9 \text{ years}$

Real-World Application: Electrician — Checking Voltage Consistency

An electrician takes 30 voltage readings from a commercial power outlet to verify it stays within the acceptable range of $114\text{V}$ to $126\text{V}$ (the standard $120\text{V} \pm 5\%$):

$117.2, \, 118.5, \, 119.0, \, 119.3, \, 119.8, \, 120.1, \, 120.4, \, 118.9, \, 121.0, \, 119.5$

$120.8, \, 121.3, \, 118.2, \, 119.7, \, 120.0, \, 120.6, \, 121.5, \, 119.1, \, 118.8, \, 120.3$

$121.8, \, 122.0, \, 119.4, \, 120.2, \, 118.6, \, 120.9, \, 121.1, \, 119.6, \, 120.5, \, 121.7$

Step 1: Organize into a frequency table with 1-volt intervals.

Step 2: Interpret the results.

• All 30 readings fall within the $114\text{V}$ to $126\text{V}$ acceptable range.
• The distribution is concentrated around $119\text{V}$ to $121\text{V}$ — that’s $\frac{8 + 9 + 6}{30} = 76.7\%$ of all readings.
• The shape is roughly symmetric and centered near $120\text{V}$.

Conclusion: The power supply is consistently within spec. A histogram of this data would show a bell-shaped distribution centered near the target of $120\text{V}$, confirming stable and reliable voltage.

Frequency Table Reference

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Frequency tables organize raw data by counting how often each value or range appears
• Use class intervals (bins) when data has many distinct values — aim for 5-10 intervals of equal width
• Relative frequency converts counts to percentages, making comparison across different-sized datasets easier
• Cumulative frequency provides a running total that answers “how many values are at or below this point?”
• Histograms are bar charts for continuous data — bars touch, and the shape reveals the distribution pattern
• You can estimate the mean from a frequency table using midpoints: $\frac{\sum(\text{midpoint} \times \text{frequency})}{\sum \text{frequency}}$

Return to Statistics for more topics in this section.