SL 4.2 · Organize and display discrete and continuous data

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Statistics and Probability

Choosing tables and graphs that preserve what the data mean

What you will learn

• Distinguish discrete data from continuous data and choose a suitable display.
• Organize observations in frequency tables and represent them with appropriate graphs.
• Construct and interpret histograms, including classes with unequal widths.
• Use cumulative frequency to display grouped continuous data and estimate values from it.

1. Begin with the type of data

• Choose the display after identifying whether the observations are counts or measurements.
• Label axes with the variable and its units; include a clear title.
• A bar chart has separated bars; a histogram has adjacent bars.

2. Make frequency displays meaningful

• For unequal class widths, use frequency density as histogram height.
• Histogram bar area, rather than height alone, represents frequency.
• Cumulative frequency is a running total and cannot decrease.

3. Interpret and check your display

• Check class boundaries, totals, units, and axis scales before interpreting.
• Distinguish exact recorded counts from estimates read from grouped displays.
• Use technology to verify a reasoned method, not as a substitute for it.

Frequency information for the discrete data

Worked example

A discrete frequency table and bar chart

1. List the possible observed values
   The observations are counts, so they are discrete. The smallest recorded count is 0 and the largest is 4. Include each integer value in this range, even if its frequency is zero.
2. Count each value
   Tallying the list gives the frequency for each number of books. Check that the frequencies add to the number of students.
   $$2+3+4+2+1=12$$
3. Choose and describe the graph
   Draw a bar chart with number of books on the horizontal axis and number of students on the vertical axis. Separate the bars because the values are distinct count categories. The tallest bar is at 2 books, so that was the most common count.

Worked example

A histogram with unequal class widths

1. Find each class width
   Subtract the lower boundary from the upper boundary. The class widths are not all equal, so the frequencies themselves cannot be used as the bar heights.
   $$10,\ 20,\ 10$$
2. Calculate frequency densities
   Divide each frequency by its class width. These values are the histogram heights; their units are students per minute.
   $$\frac{8}{10}=0.8,\quad \frac{18}{20}=0.9,\quad \frac{10}{10}=1.0$$
3. Check the rectangle areas
   For each class, multiply its density by its width. The area should reproduce the class frequency, confirming that the chosen heights represent the data correctly.
   $$0.8(10)=8,\quad 0.9(20)=18,\quad 1.0(10)=10$$

Worked example

Building and using cumulative frequency

1. Calculate running totals
   Add each class frequency to the total from the preceding classes. Plot these totals against the upper class boundaries, and include the starting point at the lower boundary with cumulative frequency zero.
   $$5,\quad 5+13=18,\quad 18+15=33,\quad 33+7=40$$
2. Locate the required cumulative frequency
   About 20 runners is halfway through the 40 observations. On a cumulative frequency graph, find 20 on the vertical axis, move across to the plotted curve, then down to the time axis. This level lies in the 20–30 minute class because cumulative frequency rises from 18 to 33 there.
3. Estimate within the class
   The increase from 18 to 33 is 15 runners across 10 minutes. Reaching 20 requires an increase of 2 runners beyond 18. Assuming the runners are spread evenly within this class gives an estimated additional time of about 1.3 minutes.
   $$20+\frac{20-18}{15}\times 10\approx 21.3$$

Common mistakes and how to avoid them

Lesson summary

• Identify whether the data are discrete counts or continuous measurements before choosing a display.
• Use frequency tables to organize observations; use separated bars for discrete categories and adjacent rectangles for continuous intervals.
• With unequal histogram class widths, use frequency density so rectangle area represents frequency.
• Cumulative frequency is a running total plotted against class boundaries; values read from grouped displays are estimates.

Check your understanding

Key terms

Discrete data

Data that take separate, countable values, such as a number of items.

Continuous data

Measurements that can take any value within a range, such as time or length.

Frequency

The number of observations in a value or class.

Class width

The difference between the upper and lower boundaries of a class interval.

Frequency density

Frequency divided by class width; used as histogram height when class widths differ.

Cumulative frequency

The running total of frequencies up to a specified class boundary.

Continue through IB AA SL

View the complete IB AA SL International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL curriculum and lessons

• SL 4.1 · Distinguish populations, samples, variables, sampling methods, and bias
• SL 4.3 · Calculate and interpret measures of centre, position, and dispersion
• SL 4.4 · Analyse correlation and linear regression with appropriate caution
• SL 4.5 · Use sample spaces, events, complements, and expected frequencies
• SL 4.6 · Solve combined and conditional probability problems
• SL 4.7 · Use discrete random-variable distributions and expected value

Tutor support for SL 4.2 · Organize and display discrete and continuous data

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Published by DoAssignment. This reviewed lesson follows International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL, study topic SL 4.2. It is a study resource, not an official curriculum publication.

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