7.1 Dispersion
In Form 4, dispersion was analysed for ungrouped data using stem-and-leaf plots and dot plots. For grouped data, dispersion is observed and interpreted through graphical representations: histograms, frequency polygons, and ogives (cumulative frequency curves). To construct these graphs, raw data must first be organised into frequency tables containing class intervals, class limits, boundaries, midpoints, and cumulative frequencies. These graphical representations allow us to examine data spread, identify distribution shapes (symmetrical or skewed), and locate key statistical measures such as quartiles and percentiles.
Learning Objectives
- Construct a histogram and a frequency polygon for a set of grouped data.
- Compare and interpret the dispersions of two or more sets of grouped data based on a histogram and a frequency polygon, and make conclusions.
- Construct an ogive for a set of grouped data and determine the quartiles (\(Q_1\), \(Q_2\), \(Q_3\)) and percentiles (\(P_k\)).
Grouped Data Graphical Toolkit: Histogram, Frequency Polygon, and Ogive
See how one grouped data set is represented across the three primary graph types and which axis values are required for each.
Histogram
Horizontal axis: Class boundaries
Vertical axis: Frequency
Graph: Contiguous bars
Frequency Polygon
Horizontal axis: Midpoints
Vertical axis: Frequency
Graph: Closed line graph starting and ending at 0 frequency
Ogive
Horizontal axis: Upper boundaries
Vertical axis: Cumulative frequency
Graph: S-shaped smooth curve
Distribution Shapes
Observe whether the data is bell-shaped, uniform, right-skewed, or left-skewed.
Brief Explanation
Basic Concepts in Grouped Data Frequency Tables
Class interval is the range into which raw data is grouped. The lower limit is the smallest value, while the upper limit is the largest value in a class.
Size of Class Interval
\[\text{Size of class interval} = \frac{\text{Largest data value} - \text{Smallest data value}}{\text{Number of classes}}\]
Round up to the nearest whole number when grouping continuous or integer values. It can also be found as the difference between the lower limits of two consecutive classes, or as \(\text{Upper boundary} - \text{Lower boundary}\).
Midpoint and Boundaries
\[\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}\]
\[\text{Lower boundary} = \frac{\text{Upper limit of the class before it} + \text{Lower limit of the class}}{2}\]
\[\text{Upper boundary} = \frac{\text{Upper limit of the class} + \text{Lower limit of the class after it}}{2}\]
Cumulative Frequency
Cumulative frequency is the running total of frequencies from the first class up to the current class.
| Graph | Horizontal axis | Vertical axis | Key feature |
|---|---|---|---|
| Histogram | Class boundaries | Frequency | Contiguous bars |
| Frequency polygon | Midpoints | Frequency | Closed line graph |
| Ogive | Upper boundaries | Cumulative frequency | S-shaped smooth curve |
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
For the class interval \(150 - 154\) where the preceding class is \(145 - 149\) and the succeeding class is \(155 - 159\), calculate its midpoint, lower boundary, and upper boundary.
Why must a frequency polygon be anchored to the horizontal axis at zero frequency before the first class and after the last class?
In an ogive representing a test taken by \(N = 80\) candidates, find the cumulative frequency positions used to determine the median (\(Q_2\)) and the 60th percentile (\(P_{60}\)).
Common Mistakes
- Calculating the class size by simply subtracting the lower limit from the upper limit. For \(145 - 149\), writing \(149 - 145 = 4\) is incorrect; the actual size is \(149.5 - 144.5 = 5\) or \(150 - 145 = 5\).
- Plotting class limits or midpoints instead of class boundaries on the horizontal axis of a histogram.
- Forgetting to add the extra class intervals with zero frequency before the first class and after the last class when drawing a frequency polygon or ogive.
- Plotting cumulative frequency against midpoints or lower boundaries instead of the upper boundaries when constructing an ogive.
- Drawing the frequency polygon with curved lines instead of straight line segments, or drawing the ogive with straight line segments instead of a smooth curve.
Misconceptions
Confusing “skewed to the right” with “higher values on the right”.
A right-skewed distribution has its longest tail extending towards the higher values on the right, meaning the bulk of the data frequencies are concentrated on the left (lower values).
Believing that discrete data cannot have boundaries.
Even when raw data are integers, class boundaries (such as \(144.5\) and \(149.5\)) are used to eliminate gaps between consecutive classes so bars touch contiguously in a histogram.
Assuming percentiles and percentages are identical.
A percentile (\(P_k\)) is a data score/value below which a certain percentage (\(k\%\)) of observations fall, whereas a percentage is a fraction of a total.
Summary
- Grouped data is organised into class intervals with defined lower limits, upper limits, midpoints, and boundaries.
- The size of a class interval is found by subtracting the lower boundary from the upper boundary, or by finding the difference between lower limits of consecutive classes.
- A histogram uses contiguous bars with class boundaries on the horizontal axis and frequencies on the vertical axis.
- A frequency polygon connects midpoints of class intervals using straight lines, anchored at zero frequency before the first and after the last classes.
- Distribution shapes include symmetric (bell-shaped, uniform) and skewed (right-skewed, left-skewed).
- An ogive is an S-shaped curve plotted using upper boundaries and cumulative frequencies, used to determine quartiles (\(Q_1, Q_2, Q_3\)) and percentiles (\(P_k\)).
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 Which coordinates are plotted on the horizontal and vertical axes to construct an ogive?
2 A distribution where the frequency is concentrated at lower values and has a longer tail stretching towards higher values on the right is classified as:
B. Fill in the Blanks
3 For the class interval 60 – 69 followed by 70 – 79, the upper boundary of the class is .
4 In an ogive representing a total frequency of 200 students, the cumulative frequency used to find the third quartile \(Q_3\) is .
C. Drag and Drop: Match Each Graph or Concept
Match each statistical graph or concept to its correct textbook description. On a phone, tap the answer choice first, then tap the description box.
