The histogram and the graph
The histogram and the graph
A histogram shows a continuous quantitative variable split into classes of equal width: a bar for each class, and its height is the class’s frequency.
The heights of the 14 students of the last lesson, in classes of 10 cm:
- The bars touch each other, with no gaps: each class begins exactly where the one before ends, because the variable is continuous.
- The classes’ bounds are written under the bars’ edges, not under their middles.
- All the bars are the same width, so they are compared by their height alone.
A histogram tells how many data there are in each class, and neighbouring classes are added up.
- The class with the tallest bar is 160–170: 5 students.
- 4 students are shorter than 160 cm: .
- 5 students are 170 cm tall or more: , that is of the class.
- What cannot be told: the heights themselves. The class 150–160 holds 3 students, but the histogram does not show whether they are 151 or 159 cm tall.
A graph suits data measured again and again over time: time on the horizontal axis, and a point for each measurement. The line joining the points shows how the value changes.
The mean temperature in Jerusalem in each month, in degrees:
- Rising: from January (month 1) to July (month 7) the temperature rises.
- Constant: between July and August the line is level – the same temperature, 24 degrees.
- Falling: from August to December the temperature falls.
- Maximum: 24 degrees, in July and August. Minimum: 9 degrees, in January.
Two graphs on the same axes compare two variables over the same time. A trend is the general direction of a graph over a long time, even when it has small rises and falls.
How many students come to a school by bike and how many by car, each year:
- The bikes’ trend – rising: from 20 students in 2019 to 52 in 2025.
- The cars’ trend – falling: from 60 to 38.
- The gap between the two graphs shrank, and by 2025 it had turned round: 14 more students come by bike, .
The representation that suits depends on the kind of variable and on the question to be answered.
- A list – few data, or when the data themselves are needed.
- A frequency table – many data, and exact numbers.
- A bar chart – a qualitative or discrete quantitative variable; comparing the frequencies.
- A pie chart – parts of a whole, with few values.
- A histogram – a continuous quantitative variable in classes of equal width.
- A graph – a value that changes over time; two graphs or more – comparing them over time.
The histogram shows the weights of 40 babies born in a hospital in one week:
Which class has the most babies?
How many babies weigh less than 3 kg?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
What percent of the babies weigh 3.5 kg or more?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
20 students wrote down how many minutes they spent on homework in one day:
12, 25, 18, 31, 22, 9, 27, 15, 20, 34, 17, 23, 28, 11, 19, 26, 21, 14, 30, 24
The data are split into classes of 5 minutes: 5–10, 10–15, 15–20, 20–25, 25–30, 30–35.
Build a frequency table of the classes, and sketch the histogram.
What is the frequency of the class 20–25?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
What percent of the students spent less than 15 minutes on homework?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
The number of visitors to a museum each year, in thousands:
By how many thousand did the number of visitors fall from 2019 to 2020?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
How many thousand visitors came to the museum in the four years 2021–2024 together?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
Describe the graph: when it rises and when it falls, its maximum and minimum, and the trend in the last years.
The sales of two shops in each month of the first half of the year, in thousands of shekels:
In month 6, by how many thousand shekels are shop A’s sales greater than shop B’s?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
In which month are the two shops’ sales equal? Describe each shop’s trend.
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
Answer both questions:
- Why do the bars of a histogram touch each other, while a bar chart has gaps between them?
- In a table of running times, the last class is “8 minutes and over”. Why is its bar hard to draw in a histogram?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power