Representing Data with Graphs
Your class just voted on a field trip, and the list of 30 votes is a mess of names and tallies. A good graph shows the winner in two seconds. But the wrong graph, or a sneaky one, can tell a different story than the data. How do you pick the right graph, and how do you catch a graph that is trying to fool you?
Populations, Samples, and Variables
Before you draw anything, ask two questions: who did we collect data about, and what did we record?
Variables come in two main kinds, and the kind decides which graph fits best.
| Type of variable | What it is | Examples |
|---|---|---|
| Categorical (qualitative) | Answers are words or labels | favorite app, eye color, pet type |
| Quantitative, discrete | Answers are counted, usually whole numbers | siblings, books read, goals scored |
| Quantitative, continuous | Answers are measured and can be any value in a range | height, time, weight |
A survey has been carried out in a 7th grade class to find out which social network is preferred by the students. The following results have been obtained: Instagram,…
A survey has been carried out in a 7th grade class to find out which social network is preferred by the students. The following results have been obtained: Instagram, TikTok, TikTok, YouTube, Instagram, TikTok, YouTube, Instagram, Instagram, TikTok, YouTube, TikTok, Instagram, YouTube, Instagram.
State what the population of this study is, what the individual is and what type of statistical variable it is.
A video game company wants to know how many hours a week young people aged $12$ to $14$ play. They have asked $100$ boys and girls chosen at random from different middle…
A video game company wants to know how many hours a week young people aged $12$ to $14$ play. They have asked $100$ boys and girls chosen at random from different middle schools in the US.
State in this case:
- a) What is the population?
- b) What is the sample?
- c) What is the individual?
- d) Is it a qualitative or quantitative variable?
Frequency Tables
A frequency table organizes raw data so a graph is easy to make. Let $N$ be the total number of data values.
Why do we need relative frequency? Because "8 students" means more in a class of 20 than in a school of 2,000. A fraction or percent lets you compare groups of different sizes.
Here is the number of pets owned by 20 students: 1, 0, 2, 1, 3, 1, 2, 0, 1, 2, 1, 4, 0, 1, 3, 2, 1, 2, 3, 1.
| Pets | Frequency $n_i$ | Relative frequency $f_i$ | Percent | Cumulative $N_i$ |
|---|---|---|---|---|
| 0 | 3 | 0.15 | 15% | 3 |
| 1 | 8 | 0.40 | 40% | 11 |
| 2 | 5 | 0.25 | 25% | 16 |
| 3 | 3 | 0.15 | 15% | 19 |
| 4 | 1 | 0.05 | 5% | 20 |
A two-way table counts two categorical variables at once. Every row and column adds up to its total, so you can fill in a missing number by subtracting.
| Sport | Boys | Girls | Total |
|---|---|---|---|
| Basketball | 24 | ? | 40 |
| Soccer | ? | 18 | 35 |
| Track | 9 | 16 | 25 |
| Total | 50 | 50 | 100 |
Basketball girls: $40 - 24 = 16$. Soccer boys: $35 - 18 = 17$.
The students in a class were asked about the number of siblings they have, obtaining the following data: $1, 0, 2, 1, 1, 0, 3, 2, 1, 1, 0, 2, 1, 4, 1, 0, 2, 1, 1, 2$…
The students in a class were asked about the number of siblings they have, obtaining the following data:
$1, 0, 2, 1, 1, 0, 3, 2, 1, 1, 0, 2, 1, 4, 1, 0, 2, 1, 1, 2$
Construct a frequency table that includes the absolute frequency ($n_i$) and the relative frequency ($f_i$).
Pictographs
A pictogram (pictograph) uses small symbols, and each symbol stands for a fixed amount. Always read the key first.
The following pictogram shows the number of trees planted by the students of a middle school over four years. If each symbol represents $5$ trees, answer the questions:…
The following pictogram shows the number of trees planted by the students of a middle school over four years. If each symbol represents $5$ trees, answer the questions:
- a) How many trees were planted in year $2021$?
- b) In which year were exactly $15$ trees planted?
- c) What is the total number of trees planted over the four years?
The town council of a city wants to raise awareness about recycling. To do this, it has published a pictogram on the number of recycling bins installed in four different…
The town council of a city wants to raise awareness about recycling. To do this, it has published a pictogram on the number of recycling bins installed in four different neighborhoods.
If each ♻️ symbol is equivalent to $12$ bins:
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Neighborhood A: ♻️♻️♻️♻️
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Neighborhood B: ♻️♻️♻️
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Neighborhood C: ♻️♻️♻️♻️♻️
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Neighborhood D: ♻️♻️
- a) Construct the absolute frequency table for the number of bins in each neighborhood.
- b) What is the total number of bins installed in the town?
- c) Represent this same data using a bar chart to compare both methods of visualization.
An NGO has carried out a survey on the number of children in $50$ families in a neighborhood. The results have been lost, but we still have the pictogram where each…
An NGO has carried out a survey on the number of children in $50$ families in a neighborhood. The results have been lost, but we still have the pictogram where each house symbol represents $4$ families.
-
$0$ children: $\triangle \triangle \triangle$
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$1$ child: $\triangle \triangle \triangle \triangle \triangle$
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$2$ children: $\triangle \triangle \triangle \triangle$
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$3$ children or more: $?$
- a) Knowing that the total number of families is $50$, how many families have $3$ children or more? How would you represent this amount in the pictogram?
- b) Calculate the mean, median and mode of the distribution.
- c) If a family is chosen at random, what is the probability that they have fewer than $2$ children? Represent this situation with a pie chart.
Bar Graphs
A bar graph shows how many individuals fall in each category or each separate value. The height of each bar is its frequency. The bars are separated by gaps because the categories are distinct, and every bar has the same width.
When you want to compare two groups, use a grouped bar graph: two bars side by side for each category. Here are the sports from the two-way table:
Look at the following bar chart representing the favorite sport of a group of teenagers from a sports club: From the graph, answer: a) How many people prefer basketball?…
Look at the following bar chart representing the favorite sport of a group of teenagers from a sports club:
From the graph, answer:
- a) How many people prefer basketball?
- b) Which is the least practiced sport in this group?
- c) How many people have been surveyed in total?
In a study on the use of social networks, $40$ teenagers at a middle school were asked what their favorite platform is. The results obtained are as follows: TikTok: $18$…
In a study on the use of social networks, $40$ teenagers at a middle school were asked what their favorite platform is. The results obtained are as follows:
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TikTok: $18$ people.
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Instagram: $12$ people.
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YouTube: $6$ people.
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BeReal: $4$ people.
- a) Identify the population, the sample and the type of statistical variable.
- b) Create a frequency table that includes the absolute frequency ($n_i$), the relative frequency ($f_i$) and the percentage ($\%$) for each value.
- c) Represent the data in a bar chart.
A survey has been carried out among $400$ teenagers about their favorite weekend activity, differentiating between boys and girls. The data is collected in the following…
A survey has been carried out among $400$ teenagers about their favorite weekend activity, differentiating between boys and girls. The data is collected in the following two-way table, but some values are missing:
| Activity | Boys | Girls | Total |
|---|---|---|---|
| Cinema | $?$ | $65$ | $110$ |
| Video games | $80$ | $?$ | $?$ |
| Sport | $?$ | $55$ | $130$ |
| Total | $210$ | $?$ | $400$ |
- a) Complete all the missing values in the table.
- b) Represent the data using a grouped bar chart (one bar for boys and another for girls for each activity).
- c) What percentage of the girls prefer sports? Is it higher or lower than the percentage of boys who prefer sports?
Histograms
Some data, like height or time, is continuous. We group it into intervals of equal width and count how many values fall in each one. The graph is a histogram.
The interval $[57, 60)$ includes $57$ but not $60$. A student who is exactly $60$ in. goes in the next interval, $[60, 63)$. The interval with the tallest bar is the modal class. Here it is $[60, 63)$.
The time (in minutes) it takes $25$ students to get to school from their home has been recorded. The data has been grouped in the following interval table: | Time (min)…
The time (in minutes) it takes $25$ students to get to school from their home has been recorded. The data has been grouped in the following interval table:
| Time (min) | Absolute frequency ($n_i$) |
|---|---|
| $[0, 10)$ | $4$ |
| $[10, 20)$ | $12$ |
| $[20, 30)$ | $7$ |
| $[30, 40)$ | $2$ |
Represent this data using a histogram. (Remember to put the intervals on the horizontal axis and the frequencies on the vertical axis).
The height (in inches) of the $20$ members of a youth basketball club has been measured. As it is a continuous variable, the data has been grouped into the following…
The height (in inches) of the $20$ members of a youth basketball club has been measured. As it is a continuous variable, the data has been grouped into the following intervals:
$[160, 165), [165, 170), [170, 175), [175, 180), [180, 185]$.
The absolute frequencies are: $2, 5, 8, 3, 2$, respectively.
- a) State the midpoints ($x_i$) of each interval.
- b) Represent the data in a histogram. Explain how this graph differs from a conventional bar chart.
- c) What is the modal class?
Frequency Polygons, Midpoints, and Estimating the Mean
The midpoint of an interval is the number halfway between its endpoints:
$$x_i = \frac{\text{lower} + \text{upper}}{2}$$For $[57, 60)$ the midpoint is $\dfrac{57 + 60}{2} = 58.5$.
A frequency polygon connects the points (midpoint, frequency) with line segments. Add a point with frequency $0$ one interval before the first and one after the last, so the polygon touches the horizontal axis.
Midpoints also let us estimate the mean from grouped data. We do not know each exact value, so we pretend every value in an interval sits at its midpoint.
Sometimes a histogram has no numbers on the vertical axis. If one bar is labeled, use it as a ruler. Suppose a bar worth $12$ athletes is $3$ grid squares tall. Then each square is $12 \div 3 = 4$ athletes. Measure every other bar in squares, multiply by $4$, and add to find the total.
The number of books read by the students in a 7th grade class during the first quarter has been recorded. The data is:…
The number of books read by the students in a 7th grade class during the first quarter has been recorded. The data is:
$1, 2, 0, 3, 1, 1, 2, 0, 1, 4, 2, 1, 1, 0, 2, 3, 1, 2, 1, 0, 5, 2, 1, 2, 1$.
- a) Create the table of absolute and cumulative absolute frequencies ($N_i$).
- b) What percentage of students have read $2$ books or fewer?
- c) Represent a frequency polygon on the absolute frequency bar chart.
The reaction time (in seconds) of a group of athletes to the starting signal has been represented in the following histogram. The intervals are of the form $[a, b)$. a)…
The reaction time (in seconds) of a group of athletes to the starting signal has been represented in the following histogram. The intervals are of the form $[a, b)$.
- a) Calculate the total number of athletes who took part in the event knowing that in the interval $[0,15, 0,20)$ there were $12$ athletes.
- b) Calculate the midpoint of each interval.
- c) Estimate the arithmetic mean of the reaction times from the data in the histogram.
- d) What percentage of athletes have a reaction time of less than $0,20$ seconds?
Pie Charts
A pie chart (circle graph) shows how a whole is split into parts. Each sector's central angle is proportional to its share of the whole. A full circle is $360^\circ$, so
$$\text{angle} = f_i \times 360^\circ = \frac{n_i}{N} \times 360^\circ$$Why this works: if a category is $\frac14$ of the data, it gets $\frac14$ of the circle, which is $90^\circ$.
If you have counts instead of percents, use the count formula. Suppose $18$ of $40$ students ride the bus. Then $\dfrac{18}{40} \times 360^\circ = 162^\circ$.
To represent the results of a survey on favorite ice cream flavor in a pie chart, we need to calculate the angles of each section. If we know that $10$ people out of a…
To represent the results of a survey on favorite ice cream flavor in a pie chart, we need to calculate the angles of each section. If we know that $10$ people out of a total of $40$ like chocolate, what angle in degrees ($^\circ$) will the circular sector corresponding to chocolate have?
A video game company wants to represent the type of console preferred by users at a gaming fair in a pie chart. The data is: PlayStation: $45\%$ Nintendo Switch: $30\%$…
A video game company wants to represent the type of console preferred by users at a gaming fair in a pie chart. The data is:
-
PlayStation: $45\%$
-
Nintendo Switch: $30\%$
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Xbox: $15\%$
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PC: $10\%$
- a) Calculate the degrees corresponding to each circular sector using direct proportion or the angle formula: $\alpha = \frac{360^\circ \cdot n_i}{N}$.
- b) Draw an approximate pie chart using the calculated angles.
A survey has been carried out among $120$ people about their weekly physical activity. The results are presented in this incomplete table: | Activity | Absolute Freq.…
A survey has been carried out among $120$ people about their weekly physical activity. The results are presented in this incomplete table:
| Activity | Absolute Freq. ($n_i$) | Relative Freq. ($f_i$) | Sector Angle ($\alpha$) |
|---|---|---|---|
| Nothing (Sedentary) | $30$ | ? | ? |
| 1-2 days | ? | $0,25$ | ? |
| 3-5 days | $42$ | ? | ? |
| More than 5 days | ? | ? | $54^\circ$ |
- a) Complete all the gaps in the table knowing that the total number of individuals ($N$) is $120$.
- b) Represent the data in a pie chart.
- c) State what the mode of this study is.
Working Backward: Missing Values
Many problems give you only part of a table or graph. The key is that three numbers describe the same row:
$$\frac{n_i}{N} = f_i = \frac{\text{central angle}}{360^\circ}$$Find any one of them and you can find the other two. Also, the $n_i$ add up to $N$, the $f_i$ add up to $1$, and the angles add up to $360^\circ$.
A survey of $N = 80$ students about daily screen time has these results. Fill in each row:
| Screen time | $n_i$ | $f_i$ | Central angle |
|---|---|---|---|
| Under 1 h | 16 | ? | ? |
| 1 to 2 h | ? | 0.35 | ? |
| 2 to 3 h | ? | ? | $90^\circ$ |
| Over 3 h | ? | ? | ? |
A cumulative frequency bar graph works the same way. Each bar shows a running total, so every bar is at least as tall as the one before. To recover the regular frequencies, subtract neighbors: $n_i = N_i - N_{i-1}$. The last bar is the whole sample, $N$.
Reading it: $N = 20$ students. The frequencies are $4$, $9 - 4 = 5$, $15 - 9 = 6$, and $20 - 15 = 5$. The mode is $2$ books (the biggest frequency, $6$). For the median, take the 10th and 11th values. The cumulative frequency passes $10$ in the "2 books" bar, so the median is $2$.
An e-sports club has carried out a study on the weekly hours its $120$ members spend training. We have the following partial data from the frequency table, where the…
An e-sports club has carried out a study on the weekly hours its $120$ members spend training. We have the following partial data from the frequency table, where the data has been grouped into intervals of the same width:
| Training hours | Absolute frequency ($n_i$) | Relative frequency ($f_i$) | Sector angle |
|---|---|---|---|
| $[0, 5)$ | $18$ | $?$ | $?$ |
| $[5, 10)$ | $?$ | $0,25$ | $?$ |
| $[10, 15)$ | $?$ | $?$ | $144^\circ$ |
| $[15, 20]$ | $?$ | $?$ | $?$ |
Complete the frequency table and represent the data using a histogram and a superimposed frequency polygon. State what percentage of members train for $10$ hours or more a week.
The town council of a city has collected data on the recycling of domestic organic waste from $200$ families. The following pie chart shows the distribution, but the…
The town council of a city has collected data on the recycling of domestic organic waste from $200$ families. The following pie chart shows the distribution, but the data for the families who recycle "A lot" has been erased.
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Nothing ($0$ pounds/month): $15\%$
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A little ($1-5$ pounds/month): $72^\circ$ (sector angle)
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Average ($6-10$ pounds/month): $90$ families
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A lot ($>10$ pounds/month): $?$
- a) Calculate the number of families belonging to each category.
- b) Calculate the angle in degrees of the sector corresponding to the families who recycle "A lot".
- c) Create a table of absolute and relative frequencies with all the data.
In a study on the use of the school library, the number of books borrowed by a sample of 7th grade students during a quarter has been recorded. The results are shown in…
In a study on the use of the school library, the number of books borrowed by a sample of 7th grade students during a quarter has been recorded. The results are shown in the following cumulative frequency bar chart ($N_i$):
- a) How many students make up the sample?
- b) Construct the absolute frequency table ($n_i$) from the cumulative frequencies.
- c) What is the mode and the median of the number of books borrowed?
- d) Represent the data using a pie chart.
Line Graphs and Combined Graphs
A line graph shows how something changes over time. Time goes on the horizontal axis, the measurement goes on the vertical axis, and we connect the points with line segments. A segment that goes up means an increase, and a steeper segment means a faster change.
The high was $72$°F on Monday, then $74$, $78$, $76$, $82$, and $90$. The only drop was Wednesday to Thursday. The biggest rise was Friday to Saturday, $+8$°F.
A combined graph puts two kinds of data on one horizontal axis. A climate graph uses bars for rainfall (left scale) and a line for temperature (right scale). Because the two vertical scales are different, always check which scale belongs to which.
From September to November, rainfall goes up ($1.5$, $3.5$, then $5.5$ in.) while the temperature goes down ($68$°F, $58$°F, then $50$°F). The two trends move in opposite directions.
The following line graph shows the evolution of the maximum temperature in a city during the first week of May: Looking at the graph: a) On which day of the week was the…
The following line graph shows the evolution of the maximum temperature in a city during the first week of May:
Looking at the graph:
- a) On which day of the week was the highest maximum temperature recorded?
- b) Between which two consecutive days was there the greatest drop in temperature?
- c) What was the maximum temperature on Monday?
A climate graph is a graph that combines a bar chart (rainfall) and a line graph (temperatures). Look at the simplified data for a quarter in a city in Southern…
A climate graph is a graph that combines a bar chart (rainfall) and a line graph (temperatures). Look at the simplified data for a quarter in a city in Southern California:
-
January: $10^\circ C$, $60\text{ mm}$
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February: $12^\circ C$, $45\text{ mm}$
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March: $15^\circ C$, $30\text{ mm}$
- a) Draw the climate graph placing the months on the $X$ axis, the rainfall on the left $Y$ axis (bars) and the temperatures on the right $Y$ axis (line).
- b) Analyzing the graph, what trend do you observe in the transition from winter to spring?
Graphs That Mislead
The same data can look calm or dramatic, depending on how the vertical axis is drawn.
Here is a checklist for suspicious graphs:
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The vertical axis does not start at $0$ (especially on bar graphs).
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The scale is stretched or squished to exaggerate or hide a change.
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The steps on an axis are uneven.
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The axis has no numbers or labels.
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One picture or bar is drawn bigger than its value deserves.
Look at the following two graphs showing the evolution of the average temperature of a city during a spring morning. Both represent the same data, but they have been…
Look at the following two graphs showing the evolution of the average temperature of a city during a spring morning. Both represent the same data, but they have been drawn with different scales on the vertical axis.
- a) Explain why Graph B gives a feeling of much faster growth than Graph A if the data is identical.
- b) Which of the two graphs seems more objective to you for informing the public? Justify your answer based on the scale of the y-axis.
An influencer wants to show the growth of their followers on two different social networks over the last six months. To do this, they present the following graphs. a)…
An influencer wants to show the growth of their followers on two different social networks over the last six months. To do this, they present the following graphs.
- a) Explain why the graph for Social Network Y might be misleading for their sponsors if they do not look at the y-axis.
- b) Knowing that on Social Network X they started with $10.000$ followers and ended with $12.000$, and on Social Network Y they started with $50.000$ and ended with $51.000$, draw a single comparative line graph using a suitable scale that allows the real growth of both accounts to be compared.
Box Plots
A box plot (box-and-whisker plot) shows the spread of numeric data using five numbers, called the five-number summary: the minimum, the first quartile $Q_1$, the median, the third quartile $Q_3$, and the maximum.
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The median splits the data in half. $Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half.
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Each of the four parts (whisker, box left of the median, box right of the median, whisker) holds about $25\%$ of the data.
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The interquartile range is $IQR = Q_3 - Q_1$. It is the length of the box and measures the spread of the middle $50\%$.
Now compare two bus routes. The box plots show the minutes each route takes to get to school.
Route 5: min $10$, $Q_1 = 14$, median $18$, $Q_3 = 22$, max $30$. Route 9: min $8$, $Q_1 = 12$, median $19$, $Q_3 = 28$, max $34$.
The medians are almost the same ($18$ and $19$ min). The IQRs are very different: Route 5 has $22 - 14 = 8$ and Route 9 has $28 - 12 = 16$. So Route 9 is much less predictable.
Which route gets you there in $18$ minutes or less more often? For Route 5, $18$ is the median, so about $50\%$ of trips take $18$ min or less. For Route 9, $18$ is between $Q_1 = 12$ and the median $19$, so only between $25\%$ and $50\%$ of trips are that fast. Route 5 is better for that goal.
A cell phone company analyzes the battery life of two smartphone models, A and B, after one year of use. The results are presented in the following box-and-whisker plot…
A cell phone company analyzes the battery life of two smartphone models, A and B, after one year of use. The results are presented in the following box-and-whisker plot (box plot). Look at the graph and answer:
- a) What is the median battery life for each model?
- b) If a user needs the battery to last at least $18$ hours, which model has a higher proportion of devices that meet this requirement? Justify your answer using quartiles.
- c) Which of the two models has a more dispersed distribution in the central $50\%$ of the data?
Summary
Back to the start: for the field-trip vote you would use a bar graph, since the choices are separate categories, and you would make sure the axis starts at $0$ so that no bar can fool anyone. Choosing the right graph and checking its scale answers both questions from the beginning.