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Construct a histogram from a frequency table

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Step 1 of 6
Grouped frequency table: 0 to 4 visits 4 students, 5 to 9 visits 7 students, 10 to 14 visits 5 students, 15 to 19 visits 2 students Row 1: column 1 0 to 4, Frequency 4. Row 2: column 1 5 to 9, Frequency 7. Row 3: column 1 10 to 14, Frequency 5. Row 4: column 1 15 to 19, Frequency 2Frequency0 to 445 to 9710 to 14515 to 192
Students
\(\displaystyle 18\)

18 students recorded how many times they visited the library in a term. The results are grouped in the table. On a frequency histogram of these data, what is the height of the column for 10 to 14 visits?

Year 12, VCE Units 3 & 4, Chapter 12, 4 difficulty levels, Calculator allowedBuild a worksheet

Constructing a histogram

A histogram displays ordered numerical intervals with touching columns. With equal class widths, the chosen frequency or percentage-frequency scale determines each height

\[h_i=f_i\quad\text{or}\quad h_i=p_i\]
  1. Put intervals in order Place numerical values or class intervals from smallest to largest on the horizontal axis \(\displaystyle 0\text{--}4,\ 5\text{--}9,\ 10\text{--}14\)
  2. Make columns touch A histogram has no gaps between adjacent columns because it displays numerical data \(\displaystyle \boxed{\phantom{a}}\boxed{\phantom{a}}\boxed{\phantom{a}}\)
  3. Use the interval frequency For the equal-width classes used here, choose a frequency or percentage-frequency scale and use each class separately, not a running total \(\displaystyle h_i=f_i\quad\text{or}\quad h_i=p_i\)
  4. Keep the scale consistent Label the vertical axis as frequency or percentage and choose a maximum above the tallest column \(\displaystyle \text{tallest height}=5,\quad \text{axis maximum}=6\)

Worked example

The frequency table below records goals scored in a set of matches. Construct the histogram on a frequency scale

Frequency histogram with touching unit-width columns centred on goals zero through five, with heights two, four, five, three, one and one; the vertical scale is zero, two, four, six Frequencies: -0.5 to less than 0.5: 2; 0.5 to less than 1.5: 4; 1.5 to less than 2.5: 5; 2.5 to less than 3.5: 3; 3.5 to less than 4.5: 1; 4.5 to less than 5.5: 1.0246012345GoalsFrequency
\[\begin{array}{c|rrrrrr}\text{Goals}&0&1&2&3&4&5\\\text{Frequency}&2&4&5&3&1&1\end{array}\]
  1. \(\displaystyle n=2+4+5+3+1+1=16\)The frequencies total 16, so the table represents 16 matches
  2. \(\displaystyle h(2)=5\)The column for 2 has height 5 because the frequency at 2 is 5
  3. \(\displaystyle \begin{gathered}\text{scale values: }0,2,4,6\\\text{axis maximum}=6\end{gathered}\)Six is the first multiple of two above the tallest frequency of five
  4. \(\displaystyle 2-\frac{1}{2}=1.5,\quad 2+\frac{1}{2}=2.5\)For an ungrouped discrete histogram, the column of width one is centred on 2

Common mistakes

Leaving gaps between histogram columns
✕\(\displaystyle \text{numerical histogram: columns have gaps}\)✓\(\displaystyle \text{numerical histogram: columns touch}\)
Numerical intervals in a histogram are continuous across the axis, so adjacent columns touch
Using cumulative frequency as height
✕\(\displaystyle h(3)=2+4+5+3=14\)✓\(\displaystyle h(3)=3\)
A column height is the frequency of its own interval, not the sum of all earlier frequencies
Mixing frequency and percentage scales
✕\(\displaystyle \text{frequency scale: }h(2)=31.25\%\)✓\(\displaystyle \text{frequency scale: }h(2)=5\)
The vertical axis determines whether a height is a count or a percentage, so the table and axis must use the same scale

Difficulty levels

introUngrouped discrete data are shown on a frequency scale and the column heights are found directly
standardGrouped data use a stated interval width and percentage frequency, with an axis maximum also required
challengeThe interval scheme may differ from the table's original grouping and may include an empty interval or chart recognition
examA choice identifies the correct rendered histogram or a single height read from it

Questions about this topic

Why do histogram columns touch?+

A histogram represents numerical intervals on a continuous ordered scale, so adjacent intervals have no category gap between them

What does a histogram column height show?+

For equal-width classes, use each class frequency or percentage frequency on the stated scale. For a probability-density histogram, area rather than height represents the class proportion

How is a discrete value centred in a histogram?+

A column for an ungrouped integer value has width one and extends half a unit below and half a unit above the value

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 12: Continuous probability distributions

In this chapter

12A Continuous random variables and density functions: Validate and normalise a continuous probability density function12A Continuous random variables and density functions: Estimate probabilities from grouped continuous frequency dataGroup numerical data into class intervalsConstruct a histogram from a frequency table12A Continuous random variables and density functions: Finite events, probability assignments and complements12A Continuous random variables and density functions: Conditional probability from exact representationsEvaluate and determine function rules12B Probability density validity and normalisation: Validate and normalise a continuous probability density function12B Probability density validity and normalisation: Estimate probabilities from grouped continuous frequency dataSolve a linear equation12B Probability density validity and normalisation: Evaluate a polynomial or recover a coefficient parameterWork with sets and interval notation12B Probability density validity and normalisation: Round to decimal places and significant figuresInterval and conditional probabilities from a densityCompound events, the addition rule and mutual exclusivity12C Interval and conditional probabilities: Finite events, probability assignments and complements12C Interval and conditional probabilities: Conditional probability from exact representationsMean, variance and standard deviation of a continuous random variableMedian, quartiles and percentiles of a continuous random variableExpand and factor polynomial expressions12D Mean, variance, standard deviation and percentiles: Evaluate a polynomial or recover a coefficient parameterMove between roots, factors, polynomial rules and equationsSolve a quadratic exactly with the quadratic formula12D Mean, variance, standard deviation and percentiles: Round to decimal places and significant figuresLinear transformations and totals of independent continuous random variables