Statistical Parameters: Measures of Center and Spread
Two food trucks outside the stadium both advertise "average wait: 10 minutes." Would you line up at either one? A single number can hide a lot. To compare data sets, you need to know where the "center" is and how spread out the data are.
Types of data
Before you calculate anything, ask what kind of data you have. The type decides which statistics make sense.
To carry out a study on the consumption habits of teenagers on a social network, the profiles of $50$ students from a middle school have been analyzed. Classify the…
To carry out a study on the consumption habits of teenagers on a social network, the profiles of $50$ students from a middle school have been analyzed. Classify the following statistical variables obtained in the study as qualitative or quantitative (indicating whether they are discrete or continuous):
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a) Brand of cell phone used.
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b) Number of "likes" received on the latest post.
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c) Total time (in minutes) spent using the app during the day.
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d) Main topic of the videos watched (sports, comedy, music, etc.).
The mean
The mean is the "fair share" value: if everyone had the same amount, how much would each person get?
Change the data and watch where the mean (pink line) and median (green line) land:
A group of friends has recorded the number of hours a week they spend playing online video games. The data obtained is as follows:…
A group of friends has recorded the number of hours a week they spend playing online video games. The data obtained is as follows:
$12, 15, 9, 12, 18, 20, 12, 14, 15, 10$
Calculate the measures of central tendency: the arithmetic mean ($\bar{x}$), the median ($Me$), and the mode ($Mo$).
The median and the mode
A group of friends has recorded the number of unread text messages they have at the end of the day. The data obtained is: $$12, 15, 12, 10, 18, 12, 14, 20, 12, 15$$…
A group of friends has recorded the number of unread text messages they have at the end of the day. The data obtained is:
$$12, 15, 12, 10, 18, 12, 14, 20, 12, 15$$Calculate the mode of this distribution and briefly explain what this value represents in the context of the exercise.
Seven students have timed how long it takes them (in seconds) to solve a Rubik's cube. The recorded times are: $$45, 52, 38, 60, 41, 55, 48$$ Calculate the median of…
Seven students have timed how long it takes them (in seconds) to solve a Rubik's cube. The recorded times are:
$$45, 52, 38, 60, 41, 55, 48$$Calculate the median of these times. Remember that you must order the data first.
The height (in inches) of the six starting players of a school basketball team has been measured: $$165, 172, 168, 170, 182, 175$$ Calculate the median of the heights.…
The height (in inches) of the six starting players of a school basketball team has been measured:
$$165, 172, 168, 170, 182, 175$$Calculate the median of the heights. As there is an even number of data points, remember how you must proceed with the two central values.
Which measure should you use?
The mean uses every value, so one extreme value can drag it. The median only cares about the middle, so it barely moves. An extreme value far from the rest is called an outlier.
A young man saves different amounts of money weekly. Over the last $5$ weeks, his savings have been: $\$5, \$8, \$6, \$7, \$54\$$ a) Calculate the mean of his savings.…
A young man saves different amounts of money weekly. Over the last $5$ weeks, his savings have been:
$\$5, \$8, \$6, \$7, \$54\$$
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a) Calculate the mean of his savings.
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b) Calculate the median.
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c) Which of the two parameters do you think best represents the "normal" savings of this young man? Justify your answer by mentioning the effect of the value $\$54$.
In a survey about the favorite color of 7th-grade students, the following results have been obtained: | Color | Frequency ($n_i$) | | :--- | :---: | | Blue | $12$ | |…
In a survey about the favorite color of 7th-grade students, the following results have been obtained:
| Color | Frequency ($n_i$) |
|---|---|
| Blue | $12$ |
| Red | $8$ |
| Green | $5$ |
| Black | $5$ |
- a) What type of statistical variable is being studied?
- b) Calculate, if possible, the mean, median, and mode. If any parameter cannot be calculated, explain the reason why.
In an app development company, the salaries of the 10 employees are as follows: eight employees earn $1\,500\ \$$ a month, the assistant director earns $4\,000\ \$$ and…
In an app development company, the salaries of the 10 employees are as follows: eight employees earn $1\,500\ \$$ a month, the assistant director earns $4\,000\ \$$ and the director earns $7\,000\ \$$.
- a) Calculate the mean, median, and mode of the salaries.
- b) The director states: "The mean salary in our company is $\$2\,300$, we are very generous." An employee replies: "Most of us only earn $1\,500\ \$$." Explain, using the calculated parameters, why both are telling the truth but give such different impressions.
- c) Which of the three statistical parameters best represents the reality of what a typical worker in that company earns? Justify your choice.
Working backward with totals
The mean connects the total and the number of values:
$$\text{total} = \text{mean} \times \text{number of values}$$This lets you find a missing value, or combine groups.
Lucia has taken 4 math exams this term and her grades have been: $6,5$; $8,2$; $7,4$ and $9,1$. a) What is her current mean grade? b) If there is still one final exam…
Lucia has taken 4 math exams this term and her grades have been: $6,5$; $8,2$; $7,4$ and $9,1$.
- a) What is her current mean grade?
- b) If there is still one final exam left, what grade must Lucia get so that her final mean is exactly $8,0$?
A group of 10 friends has a mean age of exactly $14$ years. When the older brother of one of them joins the group, the mean increases to $15$ years. a) Calculate the age…
A group of 10 friends has a mean age of exactly $14$ years. When the older brother of one of them joins the group, the mean increases to $15$ years.
- a) Calculate the age of the older brother.
- b) If a $10$-year-old girl joins afterwards, what will be the new mean of the group of 12 people?
- c) Is it possible that, after the girl joins, the median of the group remains $14$? Give an example of the ages of the original 10 friends that satisfies this condition.
A well-known streamer is analyzing the views of his last 5 videos. The mean number of views for the first 4 videos is $12\,500$ people. If he wants the overall mean of…
A well-known streamer is analyzing the views of his last 5 videos. The mean number of views for the first 4 videos is $12\,500$ people. If he wants the overall mean of the 5 videos to rise to $15\,000$ views to attract a new sponsor, what is the minimum number of views he must get on his fifth video? Explain the process, reasoning whether it is possible for the median number of views to be greater than the mean in this case.
Puzzles with several measures
Some problems give you the mean, median, mode, and range and ask you to rebuild the data. Translate each clue into a fact about the ordered list.
There is a set of 5 positive integers. It is known that: The mean is $10$. The median is $12$. The mode is $15$ and it is unique. The range is $12$. Find the 5 numbers…
There is a set of 5 positive integers. It is known that:
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The mean is $10$.
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The median is $12$.
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The mode is $15$ and it is unique.
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The range is $12$.
Find the 5 numbers that make up this set. Explain the logical steps you have followed to deduce them.
Frequency tables
When a value repeats many times, a frequency table records how often each value occurs. To find the mean, each value counts as many times as its frequency:
$$\bar{x} = \frac{\sum (\text{value} \times \text{frequency})}{N}, \qquad N = \text{total frequency}$$The following table shows the number of books read by the students of a $2º$ grade class during the first term: | Books read ($x_i$) | Number of students ($f_i$) | |…
The following table shows the number of books read by the students of a $2º$ grade class during the first term:
| Books read ($x_i$) | Number of students ($f_i$) |
|---|---|
| $0$ | $4$ |
| $1$ | $10$ |
| $2$ | $7$ |
| $3$ | $4$ |
Calculate the arithmetic mean of the number of books read per student.
In a survey about the favorite ice cream flavor in the school cafeteria, the results shown in the following graph have been obtained: From the bar chart, identify the…
In a survey about the favorite ice cream flavor in the school cafeteria, the results shown in the following graph have been obtained:
From the bar chart, identify the mode and calculate the range of the frequencies of the flavors.
In a basketball practice, the number of three-pointers made by each player in 25 attempts has been recorded. The results are shown in the following graph: From the bar…
In a basketball practice, the number of three-pointers made by each player in 25 attempts has been recorded. The results are shown in the following graph:
From the bar chart:
- a) Draw up the absolute frequency table.
- b) Calculate the mean, median, and mode of the three-pointers made.
Grouped data and histograms
When data are grouped in intervals, you no longer know the exact values. The interval $[0, 20)$ means "from 0 up to, but not including, 20." Use the class mark (the midpoint of the interval) in place of every value in it. The result is an estimate.
$$\text{class mark} = \frac{\text{lower end} + \text{upper end}}{2}$$Histograms with a missing scale
If bar heights are proportional to the frequencies, the heights give a ratio. Suppose the bars have heights in the ratio $1 : 3 : 4 : 2$ and $N = 40$. That is $1 + 3 + 4 + 2 = 10$ parts, so one part is $40 \div 10 = 4$ students. The frequencies are $4, 12, 16, 8$. For a percentage, divide the frequencies you want by $N$ and convert.
Calculate the arithmetic mean and the mode of the following distribution of data grouped into intervals, which correspond to the weight (in pounds) of the members of a…
Calculate the arithmetic mean and the mode of the following distribution of data grouped into intervals, which correspond to the weight (in pounds) of the members of a judo team:
| Weight interval | Class mark ($x_i$) | Frequency ($n_i$) |
|---|---|---|
| $[50, 60)$ | $55$ | $4$ |
| $[60, 70)$ | $65$ | $9$ |
| $[70, 80)$ | $75$ | $5$ |
| $[80, 90)$ | $85$ | $2$ |
The following histogram shows the distribution of the grades obtained on a retake exam by 40 students. However, the frequency axis has been partially erased. Given that…
The following histogram shows the distribution of the grades obtained on a retake exam by 40 students. However, the frequency axis has been partially erased.
Given that the total number of students is $40$ and that the heights of the bars are proportional to the frequencies:
- a) Determine the absolute frequency of each interval.
- b) Calculate the estimated mean grade using the class marks.
- c) If a $5$ is needed to pass and we assume that grades are spread uniformly within each interval, what estimated percentage of students has failed the exam?
A survey has been carried out on the daily time (in minutes) that teenagers at a middle school spend using social media. The data has been grouped in the following…
A survey has been carried out on the daily time (in minutes) that teenagers at a middle school spend using social media. The data has been grouped in the following table:
| Time (min) | $[0, 30)$ | $[30, 60)$ | $[60, 90)$ | $[90, 120)$ |
|---|---|---|---|---|
| No. of students ($n_i$) | $12$ | $25$ | $x$ | $8$ |
Given that the mean usage time is $57$ minutes:
- a) Calculate the value of $x$ (absolute frequency of the interval $[60, 90)$).
- b) Determine the mode (modal class) and the median (median class) of the distribution.
- c) What percentage of students spends $90$ minutes or more on social media?
The range
The simplest measure of spread (how scattered the data are) is the range:
When the data come from a graph, read the highest and lowest points, then subtract.
The maximum temperatures recorded in a city during a spring week are shown in the following line graph: Given that the minimum temperature of the week was $14ºC$…
The maximum temperatures recorded in a city during a spring week are shown in the following line graph:
Given that the minimum temperature of the week was $14ºC$ (Monday) and the maximum was $22ºC$ (Saturday), calculate the range of the temperatures in that period.
Mean absolute deviation (MAD)
The range only looks at two values. The mean absolute deviation uses every value: it tells how far, on average, the data are from the mean.
Back to the food trucks: both have a mean wait of 10 minutes. Truck A's waits are $8, 9, 10, 10, 11, 12$: range $4$, MAD $= \frac{2+1+0+0+1+2}{6} = 1$ minute. Truck B's waits are $2, 5, 10, 10, 15, 18$: range $16$, MAD $\approx 4.3$ minutes. Same average, but Truck A is far more predictable. Some problems call the MAD the mean deviation.
A health app has recorded the daily steps (in thousands) of a teenager over the last two weeks: $8, 10, 12, 8, 9, 11, 15, 8, 10, 12, 14, 8, 9, 10$ Represent this data in…
A health app has recorded the daily steps (in thousands) of a teenager over the last two weeks:
$8, 10, 12, 8, 9, 11, 15, 8, 10, 12, 14, 8, 9, 10$
Represent this data in a frequency table and calculate:
- a) The range of the distribution.
- b) The mean absolute deviation ($DM$) from the arithmetic mean.
A study has been carried out on the battery life (in hours) of two different smartphone models, $A$ and $B$, performing 10 tests with each one. Model $A$:…
A study has been carried out on the battery life (in hours) of two different smartphone models, $A$ and $B$, performing 10 tests with each one.
Model $A$: $14, 15, 16, 14, 15, 16, 15, 14, 15, 16$
Model $B$: $10, 20, 15, 12, 18, 11, 19, 15, 14, 16$
- a) Calculate the mean of both models. What do you observe?
- b) Calculate the range and mean deviation of each model.
- c) If you had to buy one based on the reliability of the battery life, which would you choose and why?
Two basketball players, Lucia and Maria, have scored the following amounts of points in their last 6 games: Lucia: $18, 22, 20, 19, 21, 20$ Maria:…
Two basketball players, Lucia and Maria, have scored the following amounts of points in their last 6 games:
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Lucia: $18, 22, 20, 19, 21, 20$
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Maria: $10, 35, 5, 40, 12, 18$
- a) Calculate the arithmetic mean of points for each player.
- b) Calculate the range and the mean absolute deviation for both.
- c) If the coach is looking for a reliable player who guarantees a consistent amount of points, even if it is not very high, who should he choose? Justify your answer by comparing their dispersion.
Quartiles and box plots
The median splits ordered data into two halves. The quartiles split each half again:
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Q1 (lower quartile): the median of the lower half.
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Q3 (upper quartile): the median of the upper half. (If the number of values is odd, leave the median out of both halves.)
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Interquartile range: $\text{IQR} = Q_3 - Q_1$. It is the spread of the middle 50% of the data, so outliers do not affect it.
The five-number summary is: minimum, $Q_1$, median, $Q_3$, maximum. A box plot (box-and-whisker plot) draws it. About 25% of the data lie in each of the four sections.
Below is a box plot representing the grades obtained by the students of a class on a retake exam: Observe the graph and identify the following measures of position: a)…
Below is a box plot representing the grades obtained by the students of a class on a retake exam:
Observe the graph and identify the following measures of position:
- a) Minimum value and maximum value.
- b) First quartile ($Q_1$), Median ($Me$), and Third quartile ($Q_3$).
- c) What percentage of students scored higher than $8$?
In an Esports competition, two teams, "Team Alpha" and "Team Beta", have obtained the following response times (in milliseconds) in a reflex test. The data is presented…
In an Esports competition, two teams, "Team Alpha" and "Team Beta", have obtained the following response times (in milliseconds) in a reflex test. The data is presented in the following box plots:
Based on the graphs:
- a) Which team has a lower median response time?
- b) Which of the two teams is more consistent in their reflexes? Justify your answer using the interquartile range.
- c) If winning the competition requires never exceeding $200$ ms, which team has better prospects?