Frequency distributions Revision Notes

A frequency distribution organises raw data to show how often each value or range of values occurs. This organisation helps summarise large datasets, making patterns easier to identify and interpret. There are two main types: ungrouped and grouped frequency distributions, each suited to different data types and analysis goals.

• Frequency distributions summarise data by counting occurrences.
• They help reveal patterns such as common values or ranges.
• Choosing the right type depends on data size and detail needed.

Ungrouped frequency distributions list each individual data value alongside its frequency — the number of times it appears in the dataset. This method is straightforward and preserves all original data details, making it ideal for small datasets or discrete data with few distinct values.

For example, consider the test scores of 10 students: 12, 15, 15, 17, 18, 18, 18, 20, 22, 22. An ungrouped frequency distribution would list each score and count:

12 occurs 1 time, 15 occurs 2 times, 17 occurs 1 time, 18 occurs 3 times, 20 occurs 1 time, 22 occurs 2 times.

• Shows exact data values and their frequencies.
• Useful for small or discrete datasets.
• No data is lost or combined.

Grouped frequency distributions organise data into intervals or classes, such as '30 to under 40'. Each class covers a range of values, and the frequency is the count of data points within that range. This approach is especially useful for large datasets or continuous data where listing every value is impractical.

For example, if we have ages of 50 people ranging from 18 to 65, grouping into intervals like 18 to under 30, 30 to under 40, 40 to under 50, etc., helps summarise the data efficiently.

Class intervals must be mutually exclusive and cover the entire data range without gaps or overlaps.

• Data is summarised into intervals or classes.
• Simplifies large or continuous datasets.
• Class intervals are defined by lower and upper boundaries, e.g., 30 to under 40.

Grouping data into intervals offers several benefits:

It reduces the complexity of large datasets, making them easier to interpret and present.

It facilitates the creation of graphical representations like histograms and frequency polygons, which rely on grouped data intervals for visual clarity and comparison purposes.

• Simplifies data analysis and presentation.
• Enables effective graphical representation.
• Helps identify overall trends and patterns.

However, grouping data also has drawbacks: the exact values within each interval are lost, which means detailed information about individual data points is obscured. This loss of precision can affect analyses that require exact values or when the distribution within intervals is not uniform.

For instance, if the interval 30 to under 40 has a frequency of 15, we do not know how many data points are near 30 or near 39, which may be important in some contexts.

• Loss of detailed information about individual data points.
• Potential to hide important variations within intervals.
• Choice of interval width can affect interpretation.

The decision to use ungrouped or grouped frequency distributions depends on the dataset size, data type, and analysis purpose.

Use ungrouped distributions for small datasets or when exact values are important.

Use grouped distributions for large datasets or continuous data to summarise and visualise effectively.

• Consider dataset size and detail needed.
• Ungrouped for small, discrete data.
• Grouped for large, continuous data or when summarising is needed.

In a grouped frequency distribution, data are organised into classes or intervals. Each class covers a range of values. The class limits define the smallest and largest values that belong to that class.

The lower class limit is the smallest value included in the class, and the upper class limit is the largest value included.

For example, if a class is written as 10–19, then 10 is the lower class limit and 19 is the upper class limit. All data values from 10 up to and including 19 belong to this class.

• Class limits specify the range of values in each class.
• Lower class limit = smallest value in the class.
• Upper class limit = largest value in the class.

Class boundaries are values that separate classes without any gaps or overlaps. They are especially important when classes are continuous, meaning data can take any value within an interval.

Class boundaries lie halfway between the upper limit of one class and the lower limit of the next class.

For example, if one class is 10–19 and the next is 20–29, the boundary between them is halfway between 19 and 20, which is 19.5. So the first class boundary is 9.5 (halfway below 10) and 19.5, and the second class boundary is 19.5 and 29.5. This ensures classes cover all values continuously without gaps.

• Class boundaries separate classes with no gaps.
• They are found by averaging the upper limit of one class and the lower limit of the next.
• Used mainly for continuous data to ensure continuous coverage.

The midpoint (or class mark) is the value exactly halfway between the lower and upper class limits. It represents the centre of the class and is often used to estimate the average value of data in that class.

To calculate the midpoint, add the lower and upper class limits and divide by 2.

For example, for the class 10–19, midpoint = (10 + 19) ÷ 2 = 14.5.

• Midpoint = (Lower class limit + Upper class limit) ÷ 2.
• Represents the centre of the class interval.
• Useful for estimating averages from grouped data.

Class width is the size of the class interval. It is found by subtracting the lower class limit from the upper class limit and adding 1 if the data are discrete and classes include whole numbers.

For continuous data, class width = Upper class limit − Lower class limit.

For example, for the class 10–19, class width = 19 − 10 = 9 for continuous data. For discrete data where classes include integers 10 to 19, the width is often considered 10 because it includes all integers from 10 to 19 inclusive (19 − 10 + 1 = 10).

• Class width = Upper class limit − Lower class limit (continuous data).
• For discrete data with whole numbers, class width = Upper limit − Lower limit + 1.
• Indicates the range of values covered by the class.

Discrete data consist of separate, distinct values, often whole numbers. Classes for discrete data often include single values or ranges of whole numbers. In such cases, class limits and class boundaries may coincide or differ slightly, but gaps are less of a concern.

Continuous data can take any value within an interval, including decimals. Classes must cover all values without gaps, so class boundaries are used to ensure intervals touch without overlapping or leaving gaps.

For example, discrete data like number of children per family might have classes 0, 1, 2, 3, etc., while continuous data like heights might have classes 150.0–159.9 cm, 160.0–169.9 cm, etc., with boundaries at 149.95, 159.95, 169.95, etc.

• Discrete data classes often have whole number limits and may not require class boundaries.
• Continuous data classes use class boundaries to avoid gaps between intervals.
• Understanding data type guides how classes are defined and interpreted.

Histograms are graphical tools used to represent grouped frequency distributions visually. Each bar corresponds to a class interval, and the height and width of the bar together convey information about the frequency of data within that interval.

Unlike bar charts for categorical data, histogram bars are adjacent, reflecting continuous or grouped numerical data. The horizontal axis shows class intervals, while the vertical axis typically shows frequency or frequency density, depending on class widths.

Histograms help identify the shape, spread, and central tendency of data distributions, making them essential for data analysis.

• Bars represent class intervals of grouped data.
• Bars are adjacent, reflecting continuous data.
• Horizontal axis shows class intervals.
• Vertical axis shows frequency or frequency density.

When all class intervals have the same width, constructing a histogram is straightforward. The vertical axis is labelled 'frequency', and the height of each bar equals the frequency of that class.

Since all bars have equal width, the area of each bar is proportional to its height, so the area effectively represents frequency. This makes interpretation intuitive: taller bars mean higher frequency.

For example, if the class 10–20 has frequency 15 and the class 20–30 has frequency 10, bars of equal width are drawn with heights 15 and 10 respectively.

• Equal class widths mean vertical axis shows frequency.
• Bar height equals frequency.
• Bar area proportional to frequency.
• Interpretation based on bar height.

When class intervals have different widths, simply plotting frequency on the vertical axis can mislead. Wider intervals would have wider bars, making their area larger regardless of frequency, exaggerating their importance.

To correct this, the vertical axis shows frequency density instead of frequency. Frequency density is calculated by dividing the frequency of a class by its class width.

This adjustment ensures that the area of each bar (frequency density × class width) equals the frequency, preserving the correct representation of data distribution across unequal intervals.

• Unequal class widths distort frequency representation if vertical axis shows frequency.
• Frequency density = frequency ÷ class width.
• Bar area (height × width) equals frequency.
• Frequency density corrects for varying class widths.

Consider a grouped frequency distribution with two classes: 10–20 (width 10) with frequency 30, and 20–35 (width 15) with frequency 45.

Calculate frequency density for each class: For 10–20: frequency density = 30 ÷ 10 = 3. For 20–35: frequency density = 45 ÷ 15 = 3.

Both classes have the same frequency density, so their bars will have the same height but different widths on the histogram. The area of each bar equals the frequency: 3 × 10 = 30 and 3 × 15 = 45 respectively, matching the original frequencies.

• Class 10–20: frequency density = 30 ÷ 10 = 3.
• Class 20–35: frequency density = 45 ÷ 15 = 3.
• Bar heights equal frequency densities.
• Bar areas equal frequencies.

To construct such a histogram: draw the horizontal axis with class intervals marked to scale, ensuring bar widths match class widths.

On the vertical axis, mark frequency density values. For each class, draw a bar with width equal to the class width and height equal to the frequency density.

Interpreting the histogram involves recognising that the height of a bar shows frequency density, not frequency. The area of the bar corresponds to the frequency of that class, so larger areas indicate higher frequencies regardless of bar height alone.

• Horizontal axis scaled to class intervals.
• Vertical axis labelled 'frequency density'.
• Bar width = class width; bar height = frequency density.
• Bar area = frequency; interpret area for frequency comparison.

Histograms are effective for visualising grouped data distributions. When class widths are equal, frequency on the vertical axis suffices. When class widths differ, frequency density must be used to avoid misleading representations.

Always check class widths before plotting. Calculate frequency density by dividing frequency by class width. Label the vertical axis accordingly.

Interpret histograms by focusing on bar areas to compare frequencies, especially when class widths vary. This ensures accurate understanding of data distribution shapes and frequencies.

• Use frequency for equal class widths; frequency density for unequal widths.
• Calculate frequency density = frequency ÷ class width.
• Label vertical axis appropriately.
• Interpret bar areas as frequencies.
• Check class widths before plotting.

A frequency polygon is a line graph that represents grouped frequency data by plotting points at the midpoints of each class interval against their frequencies and connecting these points with straight lines. To construct one, first calculate the midpoint of each class interval using the formula:

Midpoint = (Lower class boundary + Upper class boundary) ÷ 2

For example, if a class interval is 30 to under 40, the midpoint is (30 + 40) ÷ 2 = 35. Plot these midpoints on the horizontal axis (x-axis) and the corresponding frequencies on the vertical axis (y-axis). To complete the polygon, add an extra point at the midpoint before the first class interval and after the last class interval with frequency zero, then connect all points with straight lines. This closes the polygon on the x-axis and provides a clear visual outline of the data distribution.

• Calculate midpoints for each class interval.
• Plot midpoints (x-axis) against frequencies (y-axis).
• Add zero-frequency points before first and after last intervals.
• Connect points with straight lines to form the polygon.

The shape of a frequency polygon reveals important information about the data distribution. Peaks (modes) indicate the most frequent class intervals. The spread shows how widely data values are distributed, while symmetry or skewness indicates whether data is balanced or biased towards one side. For example, a polygon with a single peak near the centre suggests a symmetric distribution, whereas a polygon with a longer tail on one side indicates skewness. The height of points reflects frequency, so taller peaks correspond to higher frequencies. Understanding these features helps summarise the dataset effectively.

• Identify peaks to find modes.
• Assess spread by observing width of polygon.
• Note symmetry or skewness from shape.
• Use height of points to understand frequency.

Overlaying two frequency polygons on the same axes allows direct comparison of two grouped frequency distributions. Differences in shape, central tendency, and spread become visible. For instance, if one polygon peaks at a higher midpoint, it suggests a higher typical value in that dataset. A wider polygon indicates greater variability. Comparing the polygons can reveal which dataset has higher frequencies in certain intervals or if one is more skewed. This visual comparison aids in interpreting relationships and differences between datasets.

• Plot both polygons on the same graph.
• Compare peaks to assess central tendency.
• Compare widths to assess spread.
• Observe differences in shape and skewness.
• Draw conclusions about relative distributions.

Consider two datasets showing test scores grouped in intervals:

Dataset A: 20–29 (5), 30–39 (12), 40–49 (18), 50–59 (10), 60–69 (5)

Dataset B: 20–29 (3), 30–39 (8), 40–49 (15), 50–59 (20), 60–69 (9)

• Calculate midpoints: 24.5, 34.5, 44.5, 54.5, 64.5.
• Plot midpoints vs frequencies for each dataset.
• Add zero-frequency points at 14.5 and 74.5 for closure.
• Connect points with straight lines for each dataset.
• Overlay both polygons on the same axes.
• Interpret: Dataset A peaks at 44.5, Dataset B peaks at 54.5, indicating Dataset B has higher scores.
• Dataset B shows a wider spread and higher frequencies in upper intervals, suggesting better performance overall.

When plotting frequency polygons, ensure class intervals are continuous and non-overlapping. Use consistent scales on axes for accurate comparison. For unequal class widths, frequency polygons are less suitable; histograms with frequency density are preferred. Always label axes clearly with units and class midpoints. When interpreting, consider sample size and context to avoid overgeneralisation. Comparing polygons is most effective when datasets have the same class intervals.

• Use continuous, non-overlapping class intervals.
• Maintain consistent axis scales.
• Avoid frequency polygons for unequal class widths.
• Label axes with units and midpoints.
• Interpret in context considering sample size.
• Compare only datasets with matching class intervals.

A frequency distribution summarises data by showing how many observations fall into each class or category. For example, if you record the number of students scoring within certain mark ranges, the frequency distribution tells you how many students scored in each range.

A cumulative frequency distribution, on the other hand, shows the running total of frequencies up to and including each class. It answers the question: 'How many observations have values less than or equal to this class?' This is useful for understanding the accumulation of data and for identifying medians and quartiles.

• Frequency distribution: counts per class.
• Cumulative frequency: running total of frequencies up to each class.

To create a cumulative frequency distribution from a frequency distribution, follow these steps:

1. Start with the first frequency; this is the first cumulative frequency.

2. Add the second frequency to the first cumulative frequency to get the second cumulative frequency, and so on for each subsequent class.

• Cumulative frequency for class 1 = frequency of class 1.
• Cumulative frequency for class n = cumulative frequency of class (n-1) + frequency of class n.

Consider the following frequency distribution of test scores:

| Score Range | Frequency | | 0–9 | 3 | | 10–19 | 5 | | 20–29 | 7 | | 30–39 | 4 |

To find the cumulative frequencies: For 0–9: cumulative frequency = 3 For 10–19: 3 + 5 = 8 For 20–29: 8 + 7 = 15 For 30–39: 15 + 4 = 19 Thus, the cumulative frequency distribution is: 3, 8, 15, 19.

• Start with first frequency as cumulative frequency.
• Add each frequency to previous cumulative total.

To reverse the process and find the frequency distribution from a cumulative frequency distribution:

1. The first frequency is the same as the first cumulative frequency.

2. For each subsequent class, subtract the previous cumulative frequency from the current cumulative frequency to find the frequency for that class.

• Frequency for class 1 = cumulative frequency for class 1.
• Frequency for class n = cumulative frequency for class n – cumulative frequency for class (n-1).

Given cumulative frequencies: 4, 9, 15, 20 for four classes, find the frequencies:

Class 1 frequency = 4 Class 2 frequency = 9 – 4 = 5 Class 3 frequency = 15 – 9 = 6 Class 4 frequency = 20 – 15 = 5 Hence, the frequency distribution is 4, 5, 6, 5.

• Subtract previous cumulative frequency to find class frequency.
• Check that the sum of frequencies equals the last cumulative frequency.

Being able to switch between frequency and cumulative frequency distributions is essential for data analysis. Cumulative frequencies are used to draw cumulative frequency graphs, which help estimate medians, quartiles, and percentiles. Understanding this conversion also aids in interpreting data spread and distribution shape.

Always check your calculations by ensuring the final cumulative frequency equals the total number of observations, and that frequencies are non-negative.

• Essential for constructing cumulative frequency graphs.
• Helps in calculating medians and quartiles.
• Ensures accurate data interpretation and analysis.

Cumulative frequency is the total number of observations that fall below or up to a certain value in a data set. For continuous data grouped into class intervals, it is calculated by successively adding the frequencies of each class interval from the lowest up to the current interval.

For example, if the frequency distribution of test scores is: 0–10: 3, 10–20: 7, 20–30: 10, then the cumulative frequencies are: up to 10 is 3, up to 20 is 3 + 7 = 10, and up to 30 is 10 + 10 = 20. This running total helps us understand how data accumulates across intervals.

• Cumulative frequency = sum of frequencies up to a class boundary.
• It shows how many data points lie below or within a given value range.
• Essential for constructing cumulative frequency graphs.

To draw a cumulative frequency polygon for continuous data, first calculate the cumulative frequencies for each class interval. Then plot points where the x-coordinate is the upper class boundary and the y-coordinate is the cumulative frequency at that boundary.

For example, if the cumulative frequencies at upper boundaries 10, 20, and 30 are 3, 10, and 20 respectively, plot points (10,3), (20,10), and (30,20).

Connect these points with straight lines to form the polygon. It is common to start the polygon at the lower boundary of the first class with cumulative frequency zero to close the graph at the origin.

• Calculate cumulative frequencies from grouped data.
• Plot points at upper class boundaries against cumulative frequencies.
• Connect points with straight lines to form the polygon.
• Start at zero cumulative frequency at the lower boundary of the first class.

A cumulative frequency curve is similar to the polygon but uses a smooth curve instead of straight lines to connect the plotted points. This smooth curve better represents continuous data and can make it easier to estimate values between plotted points.

The curve is drawn by plotting the same points as the polygon and then joining them with a smooth, flowing line. This curve is sometimes called an ogive.

Both polygon and curve represent the same cumulative frequency data; the choice depends on preference or clarity.

• Plot cumulative frequency points as for the polygon.
• Join points with a smooth curve instead of straight lines.
• Curve is useful for estimating intermediate values by interpolation.
• Both polygon and curve represent cumulative frequency distribution.

Cumulative frequency graphs allow estimation of key statistics such as the median, quartiles, and percentiles by locating the corresponding cumulative frequency on the vertical axis and reading the associated value on the horizontal axis.

For example, if the total number of observations is 100, the median corresponds to the 50th cumulative frequency. Find 50 on the vertical axis, draw a horizontal line to the curve or polygon, then drop a vertical line to the horizontal axis to estimate the median value.

Similarly, the lower quartile (25th percentile) and upper quartile (75th percentile) can be found at cumulative frequencies of 25 and 75 respectively for 100 observations. This graphical method uses interpolation between points for more precise estimates when class intervals are wide or frequencies vary.

• Median corresponds to 50% of total cumulative frequency.
• Quartiles correspond to 25% and 75% cumulative frequencies.
• Estimate values by reading off the graph at these cumulative frequencies.
• Interpolation improves accuracy between plotted points.

The shape of the cumulative frequency polygon or curve reveals information about the distribution of the data. A steep section indicates many data points in that range, while a gentle slope shows fewer data points.

If the curve is symmetric and S-shaped, the data is likely approximately normally distributed. A curve skewed to the right (long tail on the right) rises slowly at first and then steeply, indicating positive skewness. Conversely, a curve skewed to the left rises steeply at first and then levels off, indicating negative skewness.

This visual insight helps in understanding the nature of the data and in comparing different data sets.

• Steep slope = high frequency in that range.
• Gentle slope = low frequency in that range.
• S-shaped curve suggests symmetric distribution.
• Skewed curves indicate data asymmetry (positive or negative).