Displaying Data: Graphs and Charts
The student council surveyed $120$ students about which snack to sell at the football games. The answers are a long list of words. Nobody wants to read $120$ lines, but a good picture shows the winner at a glance. Each kind of graph fits a different kind of data, so choosing the right one matters as much as drawing it carefully.
Frequency Tables
Before you can draw anything, you organize the data. A frequency table lists each value (or category) and counts how many times it appears.
The frequency of a value, written $f$, is the number of times it shows up. Some problems call it the absolute frequency. The total number of data values is $N$, so the frequencies always add up to $N$.
The relative frequency of a value is the fraction of the data it makes up:
$$\text{relative frequency} = \frac{f}{N}$$You can write it as a fraction, a decimal, or a percent. All the relative frequencies add up to $1$ (or $100\%$).
The mode is the value with the greatest frequency. A cumulative frequency is a running total: it adds up the frequencies so far, from the first value down to the current one. It only makes sense when the values have a natural order, like hours or number of pets.
| Topping | $f$ | Relative frequency |
|---|---|---|
| Pepperoni | $10$ | $0.50$ |
| Cheese | $7$ | $0.35$ |
| Veggie | $3$ | $0.15$ |
| Total | $20$ | $1.00$ |
Veggie is $\frac{3}{20} = 0.15 = 15\%$, and $50\% + 35\% + 15\% = 100\%$. The mode is pepperoni.
You can explore how a list of data turns into counts with this tool. Change the numbers and watch what changes.
For ordered data, add a column of running totals. If the frequencies for $0, 1, 2, 3$ pets are $5, 9, 4, 2$, the cumulative frequencies are $5,\ 14,\ 18,\ 20$. Each one is the previous total plus the new frequency, and the last one must equal $N$.
Sometimes you only get the running totals and need the new amount in each step. Go backward: subtract the previous running total from the current one. From the totals $5, 14, 18, 20$, the new amounts are $5,\ 14 - 5 = 9,\ 18 - 14 = 4,\ 20 - 18 = 2$.
Chloe asked her $20$ teammates what their favorite sneaker color is. The results were: Red, Blue, Blue, White, Red, White, White, Blue, Red, White, Blue, Blue, White,…
Chloe asked her $20$ teammates what their favorite sneaker color is. The results were:
Red, Blue, Blue, White, Red, White, White, Blue, Red, White, Blue, Blue, White, Red, White, White, Blue, White, Blue, White.
- a) Organize this data into a frequency table that includes the color and the absolute frequency ($f_i$).
- b) What is the mode of this study?
- c) Draw a simple bar graph in your notebook representing these results.
A YouTuber analyzes the views of their latest video. In the first 4 hours, the cumulative views were: Hour 1: $150$ views. Hour 2: $400$ views (in total). Hour 3: $750$…
A YouTuber analyzes the views of their latest video. In the first 4 hours, the cumulative views were:
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Hour 1: $150$ views.
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Hour 2: $400$ views (in total).
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Hour 3: $750$ views (in total).
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Hour 4: $900$ views (in total).
Construct a table that differentiates between the absolute frequency (new views each hour) and the cumulative frequency. Represent the new views for each hour in a bar chart. In which hour was there the greatest success in attracting new viewers?
Pictographs
A pictograph (also called a pictogram) uses small pictures to stand for a fixed number of data values. Always include a key that tells how much each symbol is worth.
Wednesday: $25 \div 10 = 2.5$, so draw $2$ whole symbols and half of a third one.
A group of friends recorded how much fruit they eat per week. They represented the data in this pictograph, where each fruit symbol 🍎 represents $3$ pieces of fruit…
A group of friends recorded how much fruit they eat per week. They represented the data in this pictograph, where each fruit symbol 🍎 represents $3$ pieces of fruit eaten.
- a) How many pieces of fruit has Ashley eaten in total?
- b) Who has eaten exactly $6$ pieces of fruit?
- c) How many more pieces of fruit has Sarah eaten than Lucas?
In the school library, they want to visualize how many books have been borrowed each day of the week. They have decided to use a pictogram where each "book" symbol…
In the school library, they want to visualize how many books have been borrowed each day of the week. They have decided to use a pictogram where each "book" symbol represents $5$ loans.
| Day | Books borrowed |
|---|---|
| Monday | $20$ |
| Tuesday | $15$ |
| Wednesday | $25$ |
| Thursday | $10$ |
| Friday | $30$ |
Draw the corresponding pictogram in your notebook using a simple symbol to represent the books. Calculate how many symbols you need for each day.
An animal shelter has counted how many bags of food they consume per month. They have used this pictogram where each drawing of a "bone" $\square$ represents…
An animal shelter has counted how many bags of food they consume per month. They have used this pictogram where each drawing of a "bone" $\square$ represents $15 \text{ lb}$ of food.
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January: $\square \square \square \square$
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February: $\square \square \square$
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March: $\square \square \square \square \square$
Knowing that the shelter has $30$ dogs and that each dog should eat $0,75 \text{ lb}$ a day, construct a frequency table with the pounds consumed each month and explain if in any month the dogs have eaten less than they need (assume all months have 30 days).
Bar Graphs
A bar graph is used for categories (favorite color, type of pet) or for separate whole-number values (number of siblings). Each value gets a bar whose height equals its frequency. The bars are separated by gaps, and they all have the same width.
A good bar graph has a title, labels on both axes, and an evenly spaced scale that starts at $0$.
To read this graph, find the top of a bar and look left to the scale. The bars are $4, 8, 5,$ and $2$ tall. The tallest bar is the mode: $1$ pet. Adding the bars gives the total number of students: $4 + 8 + 5 + 2 = 19$. When a bar's top is between two scale marks, estimate how far it is between them.
In a $1º$ class, a survey was conducted on the students' preferred video game genre. Look at the following bar graph and answer the questions: a) Which video game genre…
In a $1º$ class, a survey was conducted on the students' preferred video game genre. Look at the following bar graph and answer the questions:
- a) Which video game genre do the students like the most (the mode)?
- b) How many students prefer adventure games?
- c) If we add up all the answers, how many students participated in the survey?
Look at this bar graph about the number of siblings of the students in a class and complete the frequency table below: | No. of siblings | Absolute frequency ($f_i$) | |…
Look at this bar graph about the number of siblings of the students in a class and complete the frequency table below:
| No. of siblings | Absolute frequency ($f_i$) |
|---|---|
| $0$ | |
| $1$ | |
| $2$ | |
| $3$ or more | |
| Total |
In a library, they counted how many books of each type were borrowed in a day. They have started to draw the bar graph, but the last column is missing. a) If we know…
In a library, they counted how many books of each type were borrowed in a day. They have started to draw the bar graph, but the last column is missing.
- a) If we know that $12$ comic books have been borrowed, up to what height should the "Comic" bar reach on the vertical axis?
- b) Which type of book has been borrowed the least so far?
- c) Is "Poetry" the mode? Justify your answer by looking at the graph.
From a Graph to a Table (and Back)
Graphs and tables carry the same information, so you can move between them in both directions. Here is a bar graph of favorite movie genres.
Horror is $\frac{2}{20} = 0.10$, and $0.45 + 0.30 + 0.15 + 0.10 = 1$.
Sometimes you go the other way: you know the relative frequencies and the total, and you need the counts.
In a $1º$ middle school class, a vote was held to choose the theme for the end-of-year party. The results were as follows: Superheroes ($8$ votes), Space ($6$ votes),…
In a $1º$ middle school class, a vote was held to choose the theme for the end-of-year party. The results were as follows: Superheroes ($8$ votes), Space ($6$ votes), Mystery ($10$ votes), and Sports ($4$ votes).
Calculate the relative frequency of each option and represent the data in a bar chart. Make sure the vertical axis is correctly scaled and labeled.
From the following bar chart about the number of pets the students in a class have, carry out the reverse process: 1. Create the frequency table (absolute and relative).…
From the following bar chart about the number of pets the students in a class have, carry out the reverse process:
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Create the frequency table (absolute and relative).
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Calculate the total number of students who participated in the survey.
A study has been carried out on the number of siblings of the students in a class. The relative frequency bar chart shows that: $20\%$ have no siblings. $50\%$ have 1…
A study has been carried out on the number of siblings of the students in a class. The relative frequency bar chart shows that:
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$20\%$ have no siblings.
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$50\%$ have 1 sibling.
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$25\%$ have 2 siblings.
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The rest have 3 siblings.
If there is a total of $40$ students in the class, construct the absolute frequency table and draw the absolute frequency bar chart. What height would the "1 sibling" bar reach on the absolute frequency chart?
Histograms and Frequency Polygons
Some data are measurements, like height, time, or distance. Instead of listing every value, we group them into intervals of equal width and count how many values fall in each one. A histogram is a bar graph for these grouped data, and its bars touch, because the intervals are consecutive.
The notation $[20, 40)$ means "from $20$ up to, but not including, $40$." A value of exactly $40$ goes in the next interval, $[40, 60)$. That way every value belongs to exactly one bar.
A frequency polygon connects the tops of the bars. With intervals, put a point at the midpoint of each interval's bar top (the midpoint of $[0, 20)$ is $10$). Then join the points with straight segments. Finish by bringing the polygon down to $0$ at the midpoint of the empty interval just before the first bar and the one just after the last bar (here $-10$ and $90$). With single values (like $2, 3, 4$ hours), plot the points $(x, f)$ and join them. The polygon shows the shape of the data: where most values are, and where they are few.
$\frac{16}{30} \approx 0.533$, so about $53.3\%$ of the students did $40$ minutes or more.
The number of hours that $20$ teenagers spend doing sports per week has been recorded: $2, 4, 3, 5, 2, 6, 4, 3, 3, 5, 4, 2, 6, 3, 4, 5, 2, 3, 4, 3$. Construct the…
The number of hours that $20$ teenagers spend doing sports per week has been recorded:
$2, 4, 3, 5, 2, 6, 4, 3, 3, 5, 4, 2, 6, 3, 4, 5, 2, 3, 4, 3$.
Construct the absolute frequency table and represent the data using a frequency polygon. What trend do you observe in the graph?
The height of the players on a youth basketball team has been measured (in $cm$). As the data is very varied, it has been grouped into intervals: $[150, 160)$: $4$…
The height of the players on a youth basketball team has been measured (in $cm$). As the data is very varied, it has been grouped into intervals:
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$[150, 160)$: $4$ players
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$[160, 170)$: $7$ players
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$[170, 180)$: $10$ players
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$[180, 190)$: $3$ players
Represent this data using a histogram. Remember that in a histogram, unlike a bar graph, the bars touch each other because the variable is continuous.
A video game company has recorded how many hours its users play per day. The data has been grouped into the following intervals: $[0, 2)$, $[2, 4)$, $[4, 6)$, and…
A video game company has recorded how many hours its users play per day. The data has been grouped into the following intervals: $[0, 2)$, $[2, 4)$, $[4, 6)$, and $[6, 8)$. The absolute frequencies are $25, 40, 15$ and $10$ respectively.
Draw the corresponding histogram. Next, draw the frequency polygon by joining the midpoints of the top of each bar. If the recommended healthy limit is less than $4$ hours, what percentage of users play more than the recommended amount?
Circle Graphs
A circle graph (also called a pie chart) shows how a whole is split into parts. Each sector (slice) is the same fraction of the circle as the category is of the data. A full circle is $360^\circ$, so:
$$\text{angle} = \frac{f}{N} \times 360^\circ$$Why does this work? If a category is half of the data, it should be half of the circle, which is $180^\circ$. The rule just scales that idea to any fraction.
Swimming: $0.2 \times 360^\circ = 72^\circ$, and $180^\circ + 108^\circ + 72^\circ = 360^\circ$. Use a protractor to draw each sector, one after another around the circle.
The following circle graph represents how the $40$ students in a grade travel to school. Knowing that half the class walks, a quarter takes the bus, and the rest go by…
The following circle graph represents how the $40$ students in a grade travel to school. Knowing that half the class walks, a quarter takes the bus, and the rest go by car:
- a) How many students walk to school?
- b) How many students go by bus?
- c) What angle in degrees corresponds to the sector of students who walk? (Remember that a full circle is $360^\circ$).
A group of friends wants to organize a trip and they have asked $40$ classmates about their preferred destination. The results are shown in the following table: |…
A group of friends wants to organize a trip and they have asked $40$ classmates about their preferred destination. The results are shown in the following table:
| Destination | Absolute Frequency ($f_i$) |
|---|---|
| Beach | $15$ |
| Mountains | $10$ |
| European City | $10$ |
| Adventure (Camping) | $5$ |
To represent this data in a pie chart, calculate the angle corresponding to each circular sector. Remember that the total sum of the angles must be $360^\circ$.
The following pie chart represents how the students of a school manage their free time. If we know that the school has $200$ students in total, find out how many…
The following pie chart represents how the students of a school manage their free time. If we know that the school has $200$ students in total, find out how many students prefer each activity based on the percentages indicated:
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Video games: $40\%$
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Sports: $30\%$
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Reading: $20\%$
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Music: $10\%$
Reproduce it in your notebook by calculating the degrees of each sector.
Circle Graph Problems: Missing Angles and Totals
Circle graph problems often hide a piece of information. You can recover it with two facts: all the angles add to $360^\circ$, and all the percents add to $100\%$. To go from an angle to a count, use
$$f = \frac{\text{angle}}{360^\circ} \times N$$Pizza: $\frac{150}{360} \times 72 = 30$. Pasta: $\frac{120}{360} \times 72 = 24$. Check: $30 + 18 + 24 = 72$.
When the information comes in mixed forms (a percent, a count, and a decimal), turn everything into counts out of $N$. Then the "rest" is whatever is left after subtracting.
In a survey of $120$ 6th grade students about their favorite streaming platform, the following results were obtained: $40\%$ prefer Netflix, $30$ students choose…
In a survey of $120$ 6th grade students about their favorite streaming platform, the following results were obtained: $40\%$ prefer Netflix, $30$ students choose Disney+, the relative frequency of HBO is $0,15$, and the rest opt for Twitch.
Calculate the absolute frequency of each category and draw a pie chart. To do this, you must indicate the exact angle in degrees of each circular sector using the formula: $Ángulo = \frac{f_i}{N} \cdot 360^\circ$.
Below is a pie chart of the results of a vote to choose the color of the graduation T-shirt (Blue, Red, Green). The Blue sector measures $144^\circ$ and the Red sector…
Below is a pie chart of the results of a vote to choose the color of the graduation T-shirt (Blue, Red, Green). The Blue sector measures $144^\circ$ and the Red sector measures $108^\circ$.
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Calculate the angle of the Green sector.
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If we know that $30$ people voted for the color Red, how many people participated in the vote in total?
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Calculate the absolute frequencies of the other two colors and represent the final results in a bar chart.
Line Graphs
A line graph shows how something changes over time: temperature each hour, followers each month, a plant's height each week. The horizontal axis is time, and the points are joined with straight segments.
The steeper a segment, the faster the change. A segment going up means an increase, and a segment going down means a decrease.
Some problems give temperatures in °C instead of °F. The graph is built exactly the same way: just label the vertical axis with the unit the problem uses.
From 11 AM to noon the change is $37 - 38 = -1$ (a small drop). The greatest increase, $+6$ °F, was between 9 AM and 10 AM, which is the steepest segment.
You can also draw two lines on the same graph to compare. Use different colors or styles, and the gap between the lines at any time is the difference between the two values.
Sometimes the second line is not measured but built from a rule. Say a bean sprout is $1$ in. tall at week $1$, and an "ideal" plant starts at $1$ in. and grows exactly $2$ in. every week. Start with the first value, then keep adding the same amount.
The gap is greatest in week $5$, where it is $15 - 9 = 6$ in. On the graph, that is where the two lines are farthest apart.
A dashed horizontal line is often used to mark the mean, the value you get by adding all the data and dividing by how many there are.
Houston, because its range ($4$) is smaller. Its line stays closer to its dashed mean line, while Denver's line zigzags more.
The following table shows the temperature recorded in the schoolyard over five hours on a winter morning: | Time | $8:00$ | $9:00$ | $10:00$ | $11:00$ | $12:00$ | | :---…
The following table shows the temperature recorded in the schoolyard over five hours on a winter morning:
| Time | $8:00$ | $9:00$ | $10:00$ | $11:00$ | $12:00$ |
|---|---|---|---|---|---|
| Temp. ($^\circ C$) | $4$ | $6$ | $9$ | $12$ | $11$ |
- a) Represent this data using a line graph (frequency polygon) in your notebook.
- b) In which one-hour interval did the greatest temperature increase occur?
- c) What was the temperature at ten in the morning?
A group of scientists is measuring the growth of a plant under a new LED light. Over 6 weeks, the height in inches has followed this progression: $2, 5, 9, 14, 20, 27$.…
A group of scientists is measuring the growth of a plant under a new LED light. Over 6 weeks, the height in inches has followed this progression: $2, 5, 9, 14, 20, 27$.
Represent this data in a line graph. Then, add a second line (in a different style or color) representing the "ideal height" if the plant grew exactly $4 \text{ in}$ per week (starting from the initial $2 \text{ in}$). In which week is the difference between the real and ideal plant the greatest?
Compare the following two sets of data on the mean temperature ($^\circ\text{C}$) recorded at noon over a week in two different cities: City A:…
Compare the following two sets of data on the mean temperature ($^\circ\text{C}$) recorded at noon over a week in two different cities:
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City A: $12, 14, 15, 12, 11, 13, 14$
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City B: $18, 17, 19, 18, 20, 19, 18$
Represent both cities on the same line graph (use two different colors or line styles). Calculate the mean temperature of each city and draw a dotted horizontal line representing the mean of City B. Which of the two cities has more stable temperatures (smaller range)?
Choosing a Graph and Spotting Misleading Ones
Pick the graph that matches what you want to show.
| I want to show… | Use a… |
|---|---|
| A quick, visual comparison of a few amounts | Pictograph |
| Frequencies of categories or separate values | Bar graph |
| Measurements grouped into intervals | Histogram |
| Parts of a whole | Circle graph |
| Change over time | Line graph |
Graphs can also trick you. The most common trick is a truncated axis: the scale doesn't start at $0$, so small differences look huge. Both graphs below show the same data: a sneaker store sold $50$ pairs in January and $52$ pairs in February.
On the right, February's bar looks twice as tall as January's, but sales rose by only $2$ pairs, about $4\%$. Always check where the vertical axis starts, whether the scale is evenly spaced, and whether the axes are labeled.
Back to the start: the student council wants to show which snack is the winner. A bar graph compares the snacks, and a circle graph shows what share of the $120$ students chose each one. A line graph would be a poor choice, because snacks aren't ordered in time.
Match each statistical variable with the type of graph you think is most suitable to represent it (you can repeat options): 1. Changes in a baby's height month by month.…
Match each statistical variable with the type of graph you think is most suitable to represent it (you can repeat options):
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Changes in a baby's height month by month.
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Eye color of the students in your class.
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Distribution of a city budget across different areas.
A) Circle graph.
B) Bar graph.
C) Line graph.
Look at the two graphs below. They represent the sales of a sneaker store in two different months. 1. Which of the two graphs could be misleading or give a false…
Look at the two graphs below. They represent the sales of a sneaker store in two different months.
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Which of the two graphs could be misleading or give a false impression that sales have grown enormously?
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Look at the vertical axis. Do they both start at zero? Explain why it is important that statistical axes are not "truncated."