Statistics with Technology and Critical Reading
The media, social networks, and advertising are full of graphs and percentages. Some inform well; others, without exactly lying, mislead. Knowing statistics also helps you defend yourself: to read a piece of data and ask yourself, "Is this true, or is someone trying to sell me something?"
Statistics with a Spreadsheet
A spreadsheet does in a second what takes a while by hand. The most useful functions (in Spanish):
| I want… | Function |
|---|---|
| Mean | =PROMEDIO(A1:A20) |
| Median | =MEDIANA(A1:A20) |
| Mode | =MODA(A1:A20) |
| Maximum / minimum | =MAX(…) / =MIN(…) |
| Quartiles | =CUARTIL(A1:A20, 1) |
| Mean absolute deviation | =DESVPROM(A1:A20) |
| Standard deviation (population) | =DESVEST.P(A1:A20) |
| Count how many meet a condition | =CONTAR.SI(A1:A20, 5) |
Guided Practice: Data, Calculations, Graph, and Conclusion
We continue the investigation of arrival time at school with these made-up data, in minutes: 4, 6, 8, 10, 10, 12, 14, 16. They represent a sample for learning the procedure, not real results from the school.
-
Type
Minutesin A1 and the eight data values in A2:A9, one per cell. Before calculating, estimate: the mean should be close to 10. -
In C1 type
Measureand in D1,Result. Use the table below to complete C2:D6. -
In F1 type
Minutesand in G1,Frequency. In F2:F8 enter 4, 6, 8, 10, 12, 14, 16. In G2 type=CONTAR.SI($A$2:$A$9,F2)and copy it down to G8. You should get 1, 1, 1, 2, 1, 1, 1; the sum is 8. -
Select F1:G8 and insert a column graph with separate columns. Use "Arrival time: sample of 8 students" as the title, "Minutes" on the horizontal axis and "Number of students" on the vertical axis, starting at zero. Here we are representing the recorded minutes as discrete values; with intervals we would use a histogram.
| Cell C | Cell D: formula | Expected result |
|---|---|---|
| Mean | =PROMEDIO(A2:A9) |
10 min |
| Median | =MEDIANA(A2:A9) |
10 min |
| Mean absolute deviation | =DESVPROM(A2:A9) |
3 min |
| Standard deviation of the observed data set | =DESVEST.P(A2:A9) |
Approximately 3.74 min |
| Proportion of times of at most 10 min | =CONTAR.SI(A2:A9,"<=10")/CONTAR(A2:A9) |
0.625; with percent formatting, 62.5 % |
Function names and separators may change depending on the application and its regional settings: with a decimal point (as in Mexico and the US), arguments are separated with a comma; with a decimal comma, with a semicolon. Look up the equivalent function by its description if your spreadsheet uses different names.
Working as a Team and Learning from Mistakes
In a team of three, divide the roles of recorder, checker, and spokesperson, and rotate them whenever you change the data. Before deciding, each person explains their proposal; the team records one reason in favor and one limitation. If there is disagreement, go back to the data or try both options. Question the argument without putting down the person, and make sure everyone can take part, including those who need more time or an enlarged table.
If you get stuck, write down what you feel and which specific step you don't understand. Try with less data, draw a table, or ask someone to explain their strategy. When you finish, complete: "I thought that…; when I checked…; now I would change…". Correcting a formula or switching methods is part of mathematical work.
The figure shows a spreadsheet with the ages of the members of a book club. In cells B11 and B12 the sheet shows the results of the formulas $=\text{AVERAGE}(B2:B9)$ and…
The figure shows a spreadsheet with the ages of the members of a book club. In cells B11 and B12 the sheet shows the results of the formulas $=\text{AVERAGE}(B2:B9)$ and $=\text{MEDIAN}(B2:B9)$, respectively. The top bar shows the formula of cell B11, which is the selected cell.
- a) What does the formula in cell B11, $=\text{AVERAGE}(B2:B9)$, calculate?
- b) And the one in cell B12, $=\text{MEDIAN}(B2:B9)$?
- c) Calculate both values by hand from the ages in B2 to B9 and check that they match the ones the sheet shows in B11 and B12.
A statistical calculator indicates that the mean of the $10$ marks of a group is $6.4$ and the median is $6.5$. a) If the teacher decides to give $0.5$ extra marks to…
A statistical calculator indicates that the mean of the $10$ marks of a group is $6.4$ and the median is $6.5$.
- a) If the teacher decides to give $0.5$ extra marks to all students, what will the new mean and the new median be? Reason your answer without knowing the individual marks.
- b) If instead they multiply all the marks by $1.1$, what happens to the mean and the median?
- c) The sum of the $10$ marks was $64$. After checking, a mark of $4$ is corrected to $7$. What is the new mean?
The figure shows a spreadsheet with the points per match scored by a basketball player, as shown below. a) What value will cell B10 show if it contains the formula…
The figure shows a spreadsheet with the points per match scored by a basketball player, as shown below.
- a) What value will cell B10 show if it contains the formula $=\text{PROMEDIO}(B2:B8)$? Calculate it.
- b) Cell B11 contains $=\text{MAX}(B2:B8) - \text{MIN}(B2:B8)$. Which statistical parameter does it calculate and what is its value?
- c) Write in your own words the formula that should be entered to obtain the number of matches in which she scored more than $15$ points.
Graphs That Mislead
In a bar graph, cutting off the vertical axis can exaggerate a small difference, because we interpret the length of the bars as quantity. Other graphs, such as line graphs showing change over time, can use an axis that doesn't start at zero if the scale is clearly marked: you have to judge what is being represented and how.
Other common tricks:
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Pie charts that don't add up to 100 %, or tilted 3D graphs where the slice in front looks bigger.
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Pictograms in which a figure twice as tall is also twice as wide: it takes up 4 times more space!
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Different scales on two graphs that are being compared.
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Small or biased samples: "9 out of 10 users recommend it" (out of how many? who chose them?).
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Percentages without the base: "sales go up 100 % during El Buen Fin" (from 2 to 4 units).
A television channel showed this bar chart about the audience of two programs, as shown in the figure. a) At first glance, how many times more audience does program B…
A television channel showed this bar chart about the audience of two programs, as shown in the figure.
- a) At first glance, how many times more audience does program B seem to have than A?
- b) Read the actual values from the bars and calculate the true proportion between both audiences.
- c) What trick does the graph use to exaggerate the difference? How should the vertical axis be designed so as not to be misleading?
You can do it on paper and send a photo with your answer.
A soft drinks brand published this circle graph about consumer preferences, as shown in the figure. a) Add up the percentages of all the sectors. What result do you get?…
A soft drinks brand published this circle graph about consumer preferences, as shown in the figure.
- a) Add up the percentages of all the sectors. What result do you get? Is this possible in a circle graph?
- b) Explain what might have happened in the survey to obtain these data and why a circle graph is not suitable to present them.
- c) What type of chart would you choose to represent these percentages correctly?
For each of these data sets, choose the most suitable chart (bar chart, circle graph, line graph, frequency polygon or histogram) and justify your choice: a) The…
For each of these data sets, choose the most suitable chart (bar chart, circle graph, line graph, frequency polygon or histogram) and justify your choice:
- a) The percentage of time a teenager spends sleeping, studying, on leisure and on screens.
- b) The month-by-month evolution of the average temperature of a city over a year.
- c) The math grades of a class of $28$ students.
- d) The number of pets that the students in a grade have at home.