Representing Data
A good graph can be understood at a glance; a bad one confuses (or misleads). Each type of graph is suited to a certain type of data. Choosing well is just as important as drawing well, and knowing how to read a graph (in the newspaper, on social media, on a test) is just as useful.
Bar graph
For categorical or discrete quantitative variables. Each value gets a bar whose height equals its frequency. The bars are separated.
A group of $20$ teenagers answered what their favorite film genre is. The results are: Action, Comedy, Horror, Action, Comedy, Sci-Fi, Horror, Action, Action, Comedy,…
A group of $20$ teenagers answered what their favorite film genre is. The results are:
Action, Comedy, Horror, Action, Comedy, Sci-Fi, Horror, Action, Action, Comedy, Horror, Sci-Fi, Action, Comedy, Horror, Action, Action, Sci-Fi, Comedy, Action.
Create a frequency table that includes the absolute frequency ($n_i$) and draw a bar chart to represent these data clearly.
Complete the following partial frequency table on the number of times a group of friends went to the cinema in the last month: | Films ($x_i$) | Absolute frequency…
Complete the following partial frequency table on the number of times a group of friends went to the cinema in the last month:
| Films ($x_i$) | Absolute frequency ($n_i$) | Relative frequency ($h_i$) | Percentage ($\%$) |
|---|---|---|---|
| $0$ | $4$ | $0.20$ | $20\%$ |
| $1$ | $8$ | ||
| $2$ | $5$ | $0.25$ | |
| $3$ | $0.15$ | $15\%$ | |
| Total | $20$ | $1.00$ | $100\%$ |
Histogram and frequency polygon
For data grouped into intervals (continuous variables). It is like a bar graph, but the bars are touching (the intervals are consecutive) and each bar's base is its interval.
With intervals of equal width, you can use the frequency as the height. If the widths are different, the height must be the frequency density: frequency divided by width, so that the area represents the frequency. For example, 10 data values in $[0, 5)$ and 10 in $[5, 15)$ need heights 2 and 1: they have the same frequency, but not the same width.
The frequency polygon connects the midpoints of the top of each bar (the class marks).
The time (in minutes) taken by $15$ students to get from home to school has been recorded. The data are as follows:…
The time (in minutes) taken by $15$ students to get from home to school has been recorded. The data are as follows:
$12, 15, 8, 22, 10, 14, 18, 25, 11, 13, 20, 9, 15, 12, 17$.
Group these data in a frequency table using class intervals of width $5$, starting at $5$. That is: $[5, 10), [10, 15), \dots$
From the following interval table, draw the corresponding frequency histogram. Make sure the bars touch each other to indicate that the variable is continuous. |…
From the following interval table, draw the corresponding frequency histogram. Make sure the bars touch each other to indicate that the variable is continuous.
| Interval | Frequency ($f_i$) |
|---|---|
| $[5, 10)$ | 2 |
| $[10, 15)$ | 5 |
| $[15, 20)$ | 8 |
| $[20, 25)$ | 6 |
| $[25, 30)$ | 3 |
You can do it on paper and upload a photo with your answer.
The backpacks of 40 students were weighed at the entrance of the school for a health study. The results have been organized in the following table: | Weight (kg) |…
The backpacks of 40 students were weighed at the entrance of the school for a health study. The results have been organized in the following table:
| Weight (kg) | $[0, 2)$ | $[2, 4)$ | $[4, 6)$ | $[6, 8)$ | $[8, 10]$ |
|---|---|---|---|---|---|
| Students ($f_i$) | $3$ | $8$ | $15$ | $10$ | $4$ |
- a) Draw the frequency histogram.
- b) On the same graph, draw the frequency polygon.
- c) How many students carry a backpack that weighs $6$ kg or more? Calculate the relative frequency of this group (the fraction of the total that it represents), expressed as a decimal.
You can do it on paper and send a photo with your answer.
Pie chart
To see what part of the total each value represents, especially for categorical variables. In Mexico it is also called a pie chart (gráfica de pastel). Each sector (slice) has an angle proportional to its frequency:
$$\text{ángulo} = h_i \times 360° = \frac{f_i}{N} \times 360°$$The following circle graph shows the distribution of the $400$ students at a school according to the extracurricular language they have chosen to study. Look at the…
The following circle graph shows the distribution of the $400$ students at a school according to the extracurricular language they have chosen to study. Look at the chart and answer:
- a) What percentage of students study German?
- b) How many students in total study English?
- c) Which is the least popular language?
A study was carried out on the mode of transport used by students at a secundaria to get to school. The results are shown in the following circle graph, but some data…
A study was carried out on the mode of transport used by students at a secundaria to get to school. The results are shown in the following circle graph, but some data have been erased:
-
The "Walking" sector has an angle of $144^\circ$.
-
The "Bus" sector represents $25\%$ of the students.
-
The rest of the students travel by "Bicycle".
- a) Calculate the percentage of students who walk.
- b) Determine the angle in degrees corresponding to the "Bus" sector and the "Bicycle" sector.
- c) If there are 600 students in the school, how many travel by each mode of transport?
Line graph and how to choose
The line graph shows how something changes over time (followers per month, temperature per day).
| I want to show… | Graph |
|---|---|
| Frequencies of individual values | Bar graph |
| Data in intervals | Histogram (and polygon) |
| Parts of a total | Pie chart |
| Change over time | Line graph |
An "influencer" has recorded the number of new followers gained each day over a week: Monday:* $120$, Tuesday:* $150$, Wednesday:* $130$, Thursday:* $180$, Friday:*…
An "influencer" has recorded the number of new followers gained each day over a week:
Monday: $120$, Tuesday: $150$, Wednesday: $130$, Thursday: $180$, Friday: $210$, Saturday: $250$, Sunday: $230$.
Represent this trend using a frequency polygon. To do so, place the points on a coordinate axis (Day, Followers) and join them with straight line segments.
State which type of statistical graph is the most suitable (bar chart, histogram or circle graph) to represent the following situations: a) We want to compare the…
State which type of statistical graph is the most suitable (bar chart, histogram or circle graph) to represent the following situations:
- a) We want to compare the percentage of glass, paper and plastic recycling in a neighborhood.
- b) We want to show the number of absences for each student in a class (from $0$ to $5$ absences).
- c) We want to represent the weights of newborn babies in a hospital over a month (data grouped into intervals).
- d) We want to compare the preferences for $4$ different brands of soft drinks among young people in a town.
An influencer analyzes the growth of their followers on two different social networks over the last 5 months. The data (in thousands of new followers per month) are:…
An influencer analyzes the growth of their followers on two different social networks over the last 5 months. The data (in thousands of new followers per month) are:
-
Network A: $2, 5, 8, 12, 15$
-
Network B: $6, 7, 6, 8, 7$
- a) Represent both series of data on the same line graph (frequency polygon). Use a different color for each network or a dashed line for one of them.
- b) Explain, based on the shape of the graphs, which of the two social networks has more stable growth and which has a clearer upward trend.