What Is a Frequency Distribution?

Imagine recording the heights of 100 students.

A list containing 100 individual measurements would be difficult to:

read and interpret.

Instead, we can organize the measurements into groups such as:

150–159 cm
160–169 cm
170–179 cm
180–189 cm

Then we count how many observations fall into each group.

This creates a:

frequency distribution.

A frequency distribution organizes data so that we can more easily identify:

patterns, concentrations, and differences.

---

What Is Frequency?

Frequency means:

how often something occurs.

Suppose these are the numbers of books read by 10 students:

2, 3, 2, 5, 4, 3, 2, 4, 3, 2

The value 2 occurs:

4 times.

Therefore:

Frequency of 2 = 4

The value 3 occurs:

3 times.

Therefore:

Frequency of 3 = 3

---

Frequency Tables

A frequency table organizes values and shows how often each occurs.

For our book data:

This is useful when the number of possible values is:

small.

For larger datasets, we often organize values into:

groups or intervals.

---

Grouped Data

Grouped data places numerical observations into ranges called:

class intervals.

Suppose the test scores of 30 students range from:

41 to 98.

Instead of listing every individual score, we could use intervals:

40–49
50–59
60–69
70–79
80–89
90–99

We then count how many scores occur within each:

interval.

---

Class Intervals

A class interval is a range used to group numerical data.

For example:

20–29

is one class interval.

Values such as:

20, 21, 22, ... 29

belong to this group when the data are whole-number measurements and the classes are defined this way.

The next interval might be:

30–39.

Intervals should be designed so observations do not:

overlap ambiguously.

---

Why Group Data?

Grouping data makes large datasets easier to:

• organize
• summarize
• visualize
• compare
• interpret

However, grouping also removes some:

detail.

If we know that five students scored between 70 and 79, we do not know their exact scores from the grouped table alone.

This is an important trade-off:

Grouping simplifies data but sacrifices some precision.

---

Creating a Grouped Frequency Distribution

Suppose 20 students receive the following scores:

42, 47, 51, 53, 55, 58, 62, 64, 66, 67, 69, 72, 74, 75, 78, 81, 84, 86, 91, 95

We could organize them into intervals.

This table is a:

grouped frequency distribution.

---

Checking the Total Frequency

Always check that the frequencies add to the number of observations.

For our example:

2 + 4 + 5 + 4 + 3 + 2 = 20

There were:

20 students.

Therefore, all observations have been:

accounted for.

---

Using Tally Marks

When organizing raw data, tally marks can make counting easier.

As each observation is read, add a tally to the appropriate:

class interval.

Then count the tallies to determine the:

frequency.

---

Choosing Class Intervals

Good class intervals should be:

• clear
• consistent
• non-overlapping
• wide enough to summarize the data
• narrow enough to preserve useful information

Suppose ages range from:

10 to 79 years.

Using intervals of one year might produce too many:

groups.

Using only:

0–100

would produce too little information.

A reasonable choice might be:

10–19, 20–29, 30–39, and so on.

---

Class Width

The class width describes the size of an interval.

For continuous class boundaries such as:

0 to less than 10

10 to less than 20

20 to less than 30

the class width is:

10.

A consistent class width makes a frequency distribution easier to:

interpret and compare.

---

Class Boundaries

For continuous data, class boundaries help show exactly where one interval ends and another begins.

For example:

0 ≤ x < 10

means:

0 is included, but 10 is not.

The next interval could be:

10 ≤ x < 20

Now every possible value belongs to:

exactly one class.

This prevents ambiguity.

---

What Is a Histogram?

A histogram is a graph used to display the frequency distribution of:

numerical data.

It uses bars to represent how many observations fall within each:

class interval.

Histograms are especially useful for examining the:

shape of a distribution.

---

Parts of a Histogram

A histogram normally includes:

Horizontal Axis

Shows the:

numerical intervals or bins.

Vertical Axis

Usually shows:

frequency

when the class widths are equal.

Bars

Represent the frequency within each:

interval.

Title

Explains what the histogram:

represents.

Scale

Allows frequencies and intervals to be:

read accurately.

---

Histograms Have Touching Bars

One important feature of a histogram is that the bars normally:

touch.

Why?

Because the horizontal axis represents a continuous numerical scale divided into adjacent:

intervals.

For example:

0–10

followed by:

10–20

followed by:

20–30

The intervals connect continuously.

Therefore, the bars also:

touch.

---

Histogram vs Bar Graph

Histograms and bar graphs may look similar, but they represent different types of:

data.

For example:

Height intervals → Histogram

Favourite sports → Bar graph

This distinction is important.

---

Constructing a Histogram

Suppose we have the following distribution:

To construct the histogram:

Step 1: Place the time intervals on the horizontal axis.

Step 2: Place frequency on the vertical axis.

Step 3: Choose an appropriate frequency scale.

Step 4: Draw one bar for each interval.

Step 5: Make the bars touch.

Step 6: Make each bar's height equal to its frequency.

Step 7: Add axis labels and units.

Step 8: Add a descriptive title.

---

Reading a Histogram

A histogram allows us to answer questions such as:

• Which interval contains the most observations?
• Which interval contains the fewest?
• Where are most values concentrated?
• Is the distribution approximately symmetrical?
• Is the distribution skewed?
• Are there possible gaps or unusual values?
• How do two distributions differ?

A histogram therefore shows more than individual frequencies.

It reveals the:

overall shape of the data.

---

Worked Example 1: Test Scores

Suppose a class produces:

The most common interval is:

70–79.

This interval has a frequency of:

10.

The least common interval is:

90–99.

Its frequency is:

1.

The total number of students is:

2 + 5 + 8 + 10 + 4 + 1 = 30

---

The Modal Class

For grouped data, the interval with the greatest frequency is called the:

modal class.

In the previous example:

70–79

is the modal class because it has the highest frequency.

Notice that we cannot determine the exact mode from the grouped table.

We only know the interval containing the greatest number of:

observations.

---

Distribution Shape

One major reason for constructing a histogram is to examine the:

shape of a distribution.

Different datasets can produce very different shapes.

Common patterns include:

• approximately symmetrical distributions
• bell-shaped distributions
• right-skewed distributions
• left-skewed distributions
• uniform distributions
• bimodal distributions

---

Symmetrical Distributions

A distribution is symmetrical when its two sides have approximately the same:

shape.

Imagine frequencies such as:

2, 5, 9, 12, 9, 5, 2

The distribution rises toward the centre and then falls in a similar:

pattern.

The left and right sides are approximately:

mirror images.

---

Bell-Shaped Distributions

A bell-shaped distribution has:

• relatively few observations at the extremes
• many observations near the centre
• approximately symmetrical sides

Many naturally occurring measurements can sometimes show an approximately bell-shaped pattern.

However, we should not assume that every dataset is:

bell-shaped.

The shape must be determined from the:

actual data.

---

Right-Skewed Distributions

A right-skewed or positively skewed distribution has a longer tail toward the:

higher values.

Most observations occur toward the:

lower end.

A small number of unusually high values stretch the distribution to the:

right.

A common real-world example can be:

income data,

where many people may have moderate incomes while a smaller number have extremely high incomes.

---

Left-Skewed Distributions

A left-skewed or negatively skewed distribution has a longer tail toward the:

lower values.

Most observations occur toward the:

higher end.

A small number of low values stretch the distribution to the:

left.

Remember:

The direction of skew is named after the direction of the long tail.

---

Uniform Distributions

A uniform distribution occurs when frequencies are approximately similar across the:

intervals.

For example:

5, 6, 5, 5, 6, 5

would produce bars with approximately equal:

heights.

There is no strong concentration around one particular:

interval.

---

Bimodal Distributions

A bimodal distribution contains two noticeable:

peaks.

This may suggest that the data contain two different:

groups or processes.

For example, imagine measuring the heights of a mixed group containing younger children and adults.

The histogram might contain:

two clusters.

One peak could represent the children.

Another could represent the adults.

The two peaks may indicate important information about the:

population.

---

Gaps in a Distribution

A histogram may contain an interval with:

zero frequency.

This creates a gap.

For example:

The gap at:

30–39

may be meaningful.

It could indicate:

• natural separation between groups
• missing observations
• unusual sampling
• characteristics of the phenomenon being studied

A gap should therefore be:

noticed and investigated.

---

Outliers and Unusual Values

An outlier is a value that is unusually far from most of the:

other observations.

Suppose most test scores are between:

60 and 90,

but one student scores:

12.

That value may be an:

outlier.

In grouped data, histograms can sometimes suggest possible outliers, but the grouping may make it impossible to identify individual outliers precisely.

---

Worked Example 2: Reaction Times

Suppose reaction times are grouped as follows:

Most reaction times fall between:

250 ms and 349 ms.

The modal class is:

250–299 ms.

Very few observations occur at the:

extremes.

The distribution appears concentrated around the:

middle intervals.

---

Relative Frequency

Sometimes raw frequencies are difficult to compare because datasets contain different numbers of:

observations.

Suppose:

Class A has 20 students

and:

Class B has 100 students.

A frequency of 10 means very different things in each class.

For Class A:

10 out of 20 = 50%

For Class B:

10 out of 100 = 10%

To make fair comparisons, we can use:

relative frequency.

---

Calculating Relative Frequency

Use:

Relative Frequency = Frequency ÷ Total Frequency

For example:

Frequency = 8

Total = 40

Relative Frequency = 8 ÷ 40 = 0.20

This can also be expressed as:

20%

Relative frequency tells us the:

proportion of the dataset in a category or interval.

---

Relative Frequency Table

Suppose:

Relative frequencies make it easier to compare this distribution with another class containing a different number of:

students.

---

Comparing Two Frequency Distributions

Suppose two classes complete the same assessment.

Class A

Class B

Both classes contain:

20 students.

Class A is more concentrated around:

60–69.

Class B is more concentrated around:

70–89.

Therefore, the distributions differ in their:

location and shape.

---

What Should We Compare?

When comparing distributions, consider:

Centre

Where are most values concentrated?

Spread

How widely are the observations distributed?

Shape

Is the distribution symmetrical, skewed, uniform, or bimodal?

Peaks

Where are the highest frequencies?

Gaps

Are any intervals empty?

Unusual Values

Are there possible outliers?

A strong comparison describes several characteristics rather than simply saying:

"Graph A is higher."

---

Worked Example 3: Comparing Two Groups

Suppose two groups record running times.

Group A

Frequency pattern: 1, 4, 10, 4, 1

Group B

Frequency pattern: 4, 5, 3, 5, 3

Group A is strongly concentrated near the:

middle.

Group B is more:

spread out.

Even if both groups have similar central values, their distributions can still be very:

different.

This is why examining the whole distribution matters.

---

Histograms and Sample Size

A histogram containing 500 observations may look much smoother than one containing:

10 observations.

Small samples can produce irregular patterns simply because there are:

few data points.

Therefore, when comparing histograms, consider:

sample size.

A pattern based on a very small dataset may be less stable than one based on a:

large dataset.

---

Changing the Class Width

The appearance of a histogram depends partly on the:

class intervals chosen.

Suppose data range from 0 to 100.

We could group them using widths of:

5

10

or:

20.

Each choice produces a different level of:

detail.

---

Narrow Intervals

Narrow intervals provide:

more detail.

However, they can also make the histogram appear:

irregular or noisy.

Too many intervals may make the overall pattern difficult to:

see.

---

Wide Intervals

Wide intervals produce a:

simpler histogram.

However, important patterns may become:

hidden.

For example, two separate peaks might be combined into one broad:

bar.

Choosing intervals therefore requires:

judgment.

---

Same Data, Different Appearance

This is an important statistical idea.

Two histograms created from exactly the same raw data can look different if they use different:

class widths.

Therefore, when interpreting a histogram, consider not only the bars but also:

how the data were grouped.

---

Unequal Class Widths

So far, our examples have used intervals of equal width.

But consider:

0–10

10–20

20–40

40–80

These intervals have different:

widths.

If we simply use frequency as bar height, the wider intervals may appear misleadingly important because they cover a larger range.

For histograms with unequal class widths, we use:

frequency density.

---

Frequency Density

Use:

Frequency Density = Frequency ÷ Class Width

For example:

Class interval:

20–40

Frequency:

12

Class width:

20

Therefore:

Frequency Density = 12 ÷ 20 = 0.6

The height of the histogram bar should represent:

frequency density.

This ensures that the:

area of the bar

represents the frequency.

---

Why Area Matters in a Histogram

In a histogram:

Bar Area represents Frequency

When all class widths are equal, using frequency as the bar height works because every bar has the same:

width.

When class widths differ, the bar heights must be adjusted using:

frequency density.

This is a more advanced but important feature of:

histograms.

---

Worked Example 4: Unequal Intervals

Suppose:

For 20–40:

Frequency Density = 12 ÷ 20 = 0.6

For 40–80:

Frequency Density = 16 ÷ 40 = 0.4

Although the 40–80 interval has the largest frequency, its bar should not necessarily be the:

tallest.

Its interval is also much:

wider.

---

Histograms in Science

Histograms are extremely useful in scientific research.

They can show distributions of:

• plant heights
• body masses
• reaction times
• measurement errors
• particle sizes
• temperatures
• rainfall
• test results
• population characteristics

Scientists use histograms to understand:

variation within data.

---

Histograms in Quality Control

Suppose a factory produces bolts that should be:

50 mm long.

Measurements from hundreds of bolts can be displayed in a histogram.

If most bolts cluster close to:

50 mm,

the manufacturing process may be operating consistently.

If the distribution becomes wider or shifts away from 50 mm, this could indicate:

a change in the production process.

Histograms can therefore help identify:

patterns and potential problems.

---

Histograms in Education

A teacher could use a histogram to examine:

test scores.

The distribution might reveal:

• most students performed similarly
• scores were widely spread
• two different performance groups appeared
• many scores were concentrated at one end
• unusual results occurred

This provides more information than simply knowing:

the class average.

---

Histograms in Biology

Suppose a biologist measures the lengths of:

200 leaves.

A histogram could show whether most leaves are:

similar in size

or whether there is:

large variation.

If two peaks appear, the biologist might investigate whether the sample contains:

two different populations or conditions.

---

Histograms and Normal Distributions

Some datasets produce an approximately:

bell-shaped, symmetrical distribution.

This type of pattern is related to the:

normal distribution.

Many statistical methods make use of the normal distribution.

However:

Not every bell-shaped histogram is perfectly normal, and not every dataset should be expected to follow a normal distribution.

At this stage, the important skill is recognizing an approximately:

symmetrical bell-shaped pattern.

---

Interpreting Before Explaining

A histogram may show that:

Group A has a wider spread than Group B.

That is an interpretation of the:

data.

It does not automatically tell us:

why.

Perhaps the groups differ in age.

Perhaps the measurements were taken differently.

Perhaps the samples came from different populations.

Graphs reveal patterns.

Explanations require:

additional evidence.

---

Common Mistakes When Constructing Histograms

Watch for these common errors:

• leaving gaps between bars
• treating categories as numerical intervals
• overlapping class intervals
• forgetting axis labels
• forgetting measurement units
• using inconsistent scales
• plotting frequencies incorrectly
• changing class widths without accounting for frequency density
• choosing intervals that hide important patterns
• calling every bar graph a histogram

---

Evaluating a Histogram

Before accepting a histogram, ask:

Are the data numerical?

Histograms are designed for:

numerical distributions.

Are the intervals clear?

Each observation should belong to:

one interval.

Are the bars adjacent?

For continuous grouped data, the bars should:

touch.

Is the scale accurate?

The axes should represent the values:

consistently.

Are the class widths appropriate?

The intervals should reveal useful patterns without creating unnecessary:

noise.

Are unequal class widths handled correctly?

If widths differ, the graph may require:

frequency density.

---

The SHAPE Method

A useful way to interpret a histogram is to examine its:

SHAPE.

S — Symmetry or Skew

Is the distribution symmetrical or skewed?

H — High Points

Where are the peaks or modal intervals?

A — Areas of Concentration

Where are most observations located?

P — Possible Gaps or Unusual Values

Are there gaps, clusters, or possible outliers?

E — Extent

How widely are the observations spread?

This provides a systematic way to describe a:

distribution.

---

Worked Example 5: Describe the Distribution

Suppose a histogram has frequencies:

1, 3, 7, 12, 8, 4, 2

The frequencies rise toward a central peak and then:

decrease.

The distribution has one clear:

peak.

Most observations are concentrated near the:

middle.

There are relatively few observations at either:

extreme.

We could describe the distribution as:

approximately unimodal and fairly symmetrical.

---

Worked Example 6: Identify the Skew

Suppose the frequencies from low to high intervals are:

15, 11, 8, 5, 3, 1

Most observations occur among the:

lower values.

The frequencies gradually extend toward the:

higher values.

The long tail points to the:

right.

Therefore, the distribution is:

right-skewed.

---

Comparing Histograms Fairly

Suppose Histogram A contains:

50 observations

while Histogram B contains:

500 observations.

Raw frequency heights may not provide a fair comparison.

Instead, compare:

relative frequencies or percentages.

For example:

Dataset A:

20 out of 50 = 40%

Dataset B:

120 out of 500 = 24%

Although Dataset B has a larger raw frequency, Dataset A has the larger:

proportion.

---

Frequency Distributions and Data Reduction

Grouping data is a form of:

data reduction.

A large dataset is compressed into a smaller number of intervals.

This makes patterns easier to see.

However, once data have been grouped, some exact information is:

lost.

For example, if:

12 students scored between 70 and 79

we cannot determine from the grouped table whether the scores were:

70, 71, 72

or:

77, 78, 79.

This limitation should be remembered when interpreting grouped data.

---

A Complete Histogram Workflow

When creating a histogram:

Collect the numerical data

↓

Determine the data range

↓

Choose suitable class intervals

↓

Count the observations in each interval

↓

Create a frequency distribution

↓

Check the total frequency

↓

Label the horizontal axis

↓

Label the frequency axis

↓

Choose an appropriate scale

↓

Draw adjacent bars

↓

Add a descriptive title

↓

Examine the shape of the distribution

↓

Interpret the pattern

---

Check Your Understanding

1. Define frequency.

2. What is a frequency distribution?

3. What is grouped data?

4. What is a class interval?

5. Why might we group a large dataset?

6. What information is lost when data are grouped?

7. Define a histogram.

8. Why do the bars in a histogram normally touch?

9. Give two differences between a histogram and a bar graph.

10. What is the modal class?

11. Describe a symmetrical distribution.

12. What does right-skewed mean?

13. In which direction does the long tail point in a left-skewed distribution?

14. What is a bimodal distribution?

15. What might a gap in a histogram indicate?

16. Calculate the total frequency for: 3, 7, 9, 12, 6, 3.

17. An interval contains 8 of 40 observations. Calculate its relative frequency as a percentage.

18. Why are relative frequencies useful when comparing datasets of different sizes?

19. Explain how changing the class width can change the appearance of a histogram.

20. Why can very wide class intervals hide important patterns?

21. What is frequency density?

22. Calculate the frequency density if frequency = 18 and class width = 30.

23. Why is frequency density needed when class widths are unequal?

24. A histogram has two clear peaks. What term describes this distribution?

25. Describe three characteristics you should examine when comparing two frequency distributions.

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Key Terms

• Frequency: Number of times a value or observation occurs.
• Frequency distribution: Organization of data showing how observations are distributed among values or intervals.
• Frequency table: Table showing values or intervals and their frequencies.
• Grouped data: Data organized into numerical intervals.
• Class interval: Range of values used to group observations.
• Class width: Size of a class interval.
• Class boundary: Value separating one interval from another.
• Histogram: Graph displaying the distribution of grouped numerical data using adjacent bars.
• Modal class: Class interval with the greatest frequency.
• Distribution: Pattern showing how observations are spread across possible values.
• Symmetrical distribution: Distribution with approximately similar shapes on both sides.
• Bell-shaped distribution: Distribution with many observations near the centre and fewer toward the extremes.
• Skewed distribution: Distribution with a longer tail on one side.
• Right-skewed: Distribution with a long tail toward higher values.
• Left-skewed: Distribution with a long tail toward lower values.
• Uniform distribution: Distribution with approximately equal frequencies across intervals.
• Bimodal distribution: Distribution containing two noticeable peaks.
• Peak: Area containing a relatively high frequency.
• Gap: Interval containing few or no observations.
• Outlier: Observation unusually far from most other values.
• Relative frequency: Frequency expressed as a proportion of the total.
• Frequency density: Frequency divided by class width.
• Sample size: Total number of observations in a dataset.
• Data range: Difference between the highest and lowest values in a dataset.

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Key Takeaways

• Frequency describes how often a value occurs.
• A frequency distribution organizes observations according to their frequencies.
• Large numerical datasets can be organized into class intervals.
• Grouping makes patterns easier to identify but causes some exact information to be lost.
• Class intervals should be clear, consistent, and non-overlapping.
• A histogram displays the distribution of grouped numerical data.
• Histogram bars normally touch because the intervals represent adjacent portions of a numerical scale.
• Bar graphs usually compare categories, while histograms display numerical distributions.
• The interval with the greatest frequency is called the modal class.
• Histograms allow us to examine the shape of a distribution.
• A symmetrical distribution has approximately similar shapes on both sides.
• A right-skewed distribution has a long tail toward higher values.
• A left-skewed distribution has a long tail toward lower values.
• A uniform distribution has approximately similar frequencies across its intervals.
• A bimodal distribution contains two noticeable peaks.
• Gaps and unusual values may provide important information about a dataset.
• Relative frequency is calculated using: Relative Frequency = Frequency ÷ Total Frequency.
• Relative frequencies are especially useful when comparing datasets with different sample sizes.
• When comparing distributions, examine centre, spread, shape, peaks, gaps, and unusual values.
• The appearance of a histogram can change depending on the class width chosen.
• Narrow intervals reveal more detail but may create a noisy graph.
• Wide intervals simplify the graph but may hide important patterns.
• When class widths are unequal, use: Frequency Density = Frequency ÷ Class Width.
• With unequal intervals, the area of each histogram bar represents frequency.
• Histograms are widely used in science, education, manufacturing, biology, and statistics.
• A histogram can reveal a pattern but does not necessarily explain why the pattern exists.
• Effective histograms communicate the distribution and variation within a dataset clearly and accurately.