Statistical Parameters
Two cell phone models have the same average battery life: 15 hours. Are they equally reliable? A single number doesn't tell the whole story. You need to know where the "center" of the data is and also whether the data are close together or widely spread out. That is what statistical parameters measure.
Types of variables
Before calculating, you need to know what type of data you have. Which parameters you can calculate depends on it.
To carry out a study on the consumption habits of teenagers on a social network, the profiles of $50$ students from a high school were analyzed. Classify the following…
To carry out a study on the consumption habits of teenagers on a social network, the profiles of $50$ students from a high school were 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.).
Mean, median, and mode
The mean is the fair share: if everyone had the same amount, each would have $\bar{x}$. The median leaves half of the data on each side. Change the data and see where the mean and the median land:
A group of friends recorded the number of unread WhatsApp 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 recorded the number of unread WhatsApp 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 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…
Seven students 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 to order the data first.
The height (in centimeters) of the six starting players of a school basketball team was measured: $$165, 172, 168, 170, 182, 175$$ Calculate the median height. As there…
The height (in centimeters) of the six starting players of a school basketball team was measured:
$$165, 172, 168, 170, 182, 175$$Calculate the median height. As there is an even number of values, remember how to proceed with the two central values.
Outliers: mean or median?
A young person saves different amounts of money weekly. Over the last $5$ weeks, his savings have been: $$\text{RD\$}\ 300, \text{RD\$}\ 480, \text{RD\…
A young person saves different amounts of money weekly. Over the last $5$ weeks, his savings have been:
$$\text{RD\$}\ 300, \text{RD\$}\ 480, \text{RD\$}\ 360, \text{RD\$}\ 420, 3 \text{RD\$}\ 240$$-
a) Calculate the mean of his savings.
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b) Calculate the median.
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c) Which of the two measures do you think best represents the "normal" savings of this young person? Justify your answer by mentioning the effect of the value $\text{RD\$}\ 3\,240$.
In an app development company, the salaries of the 10 employees are as follows: eight employees earn $\text{RD\$}\ 90\,000…
In an app development company, the salaries of the 10 employees are as follows: eight employees earn $\text{RD\$}\ 90\,000$ per month, the deputy director earns $\text{RD\$}\ 240\,000$, and the director earns $\text{RD\$}\ 420\,000$.
- a) Calculate the mean, median, and mode of the salaries.
- b) The director claims: "The mean salary in our company is $\text{RD\$}\ 138\,000$, we are very generous." An employee responds: "Most of us only earn $\text{RD\$}\ 90\,000$." Explain, using the calculated parameters, why both are telling the truth but giving such different impressions.
- c) Which of the three statistical parameters best represents the reality of what a typical worker at that company earns? Justify your choice.
Missing data: using the sum
Since $\bar{x} = \dfrac{\sum x_i}{N}$, it follows that $\sum x_i = N \cdot \bar{x}$. With this you can find a missing value or figure out what happens when you add one.
Aisha has taken 4 Mathematics exams this term and her marks were: $6.5$; $8.2$; $7.4$ and $9.1$. a) What is her current mean mark? b) If there is still one final exam…
Aisha has taken 4 Mathematics exams this term and her marks were: $6.5$; $8.2$; $7.4$ and $9.1$.
- a) What is her current mean mark?
- b) If there is still one final exam left, what mark must Aisha get so that her final mean is exactly $8.0$?
A well-known streamer is analysing the views on their last 5 videos. The mean number of views for the first 4 videos is $12\,500$ people. If they want the total mean of…
A well-known streamer is analysing the views on their last 5 videos. The mean number of views for the first 4 videos is $12\,500$ people. If they want the total mean of the 5 videos to rise to $15\,000$ views to attract a new sponsor, how many views must they get, as a minimum, on their 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.
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 girl aged $10$ then joins, 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 10 original friends that satisfies this condition.
Parameters from a frequency table
When the data are in a table, each value is weighted by its frequency $n_i$:
$$\bar{x} = \frac{\sum x_i\, n_i}{N}, \qquad N = \sum n_i$$Ordering the 14 data values, those in the 7th and 8th positions are 6, so $Me = 6$. The mode is $5$, which has the tallest bar.
The following table shows the number of books read by the students in a $2º$ secondary school class during the first term: | Books read ($x_i$) | Number of students…
The following table shows the number of books read by the students in a $2º$ secondary school 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 basketball training session, the number of three-pointers scored by each player in 25 attempts was recorded. The results are shown in the following graph: From the…
In a basketball training session, the number of three-pointers scored by each player in 25 attempts was 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 scored.
You can do it on paper and send a photo with your answer.
Data grouped in intervals
With intervals we don't know the exact data, so we use the class mark (the midpoint of the interval) in place of $x_i$. That is why the mean is an estimate.
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Modal interval: the one with the greatest frequency.
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Median interval: the one where the cumulative frequency reaches $N/2$.
Calculate the arithmetic mean and mode of the following distribution of data grouped in intervals, which correspond to the weight (in kg) of the members of a judo team:…
Calculate the arithmetic mean and mode of the following distribution of data grouped in intervals, which correspond to the weight (in kg) of the members of a judo team:
| Weight interval | Class mark ($x_i$) | Frequency ($n_i$) |
|---|---|---|
| $[50, 60)$ | $55$ | $4$ |
| $[60, 70)$ | $65$ | $9$ |
| $[70, 80)$ | $75$ | $5$ |
| $[80, 90)$ | $85$ | $2$ |
A survey was carried out on the daily time (in minutes) spent by students at a high school using social media. The data has been grouped in the following table: | Time…
A survey was carried out on the daily time (in minutes) spent by students at a high school 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 ($n_i$) | $12$ | $25$ | $x$ | $8$ |
Knowing that the mean usage time is $57$ minutes:
- a) Calculate the value of $x$ (absolute frequency of interval $[60, 90)$).
- b) Determine the mode (modal interval) and the median (median interval) of the distribution.
- c) What percentage of students spend $90$ minutes or more on social media?
The following histogram shows the distribution of marks obtained in a resit exam by 40 students. However, the frequency axis has been partially erased. Knowing that the…
The following histogram shows the distribution of marks obtained in a resit exam by 40 students. However, the frequency axis has been partially erased.
Knowing 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 mark using class marks.
- c) If a mark of $5$ is needed to pass and we assume that marks are spread uniformly within each interval, what estimated percentage of students failed the exam?
You can do it on paper and send a photo with your answer.
The range
Both models have a mean of 15 h, but A lasts almost the same every time (range $16 - 14 = 2$) and B is very irregular (range $20 - 10 = 10$).
The maximum temperatures recorded during a week in April in Santo Domingo were: $22$, $25$, $19$, $24$, $27$, $26$ and $21$ °C. Calculate the range of the temperatures…
The maximum temperatures recorded during a week in April in Santo Domingo were: $22$, $25$, $19$, $24$, $27$, $26$ and $21$ °C. Calculate the range of the temperatures for that week.
In two $2.º$ secondary school classes, the time taken by the students to get to school was recorded. In $2.º$ A, the times range from $5$ to $40$ minutes, and in $2.º$…
In two $2.º$ secondary school classes, the time taken by the students to get to school was recorded. In $2.º$ A, the times range from $5$ to $40$ minutes, and in $2.º$ B, they range from $15$ to $25$ minutes.
a) Calculate the range of the times for each class.
b) In which class are the times most similar to each other? Why?
Two basketball players scored in their last $6$ matches: Player A: $7$, $8$, $8$, $9$, $6$, $10$ points Player B: $2$, $14$, $5$, $11$, $9$, $7$ points a) Check that…
Two basketball players scored in their last $6$ matches:
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Player A: $7$, $8$, $8$, $9$, $6$, $10$ points
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Player B: $2$, $14$, $5$, $11$, $9$, $7$ points
a) Check that both players have the same mean number of points.
b) Calculate the range of points scored by each player.
c) The coach needs a player with consistent performance for the next match. Which one would you choose? Why?
The mean deviation
The range only looks at the extremes. The mean deviation takes all the data into account: how far, on average, each value is from the mean.
With a frequency table, each deviation is multiplied by its frequency: $MD = \dfrac{\sum |x_i-\bar{x}|\, n_i}{N}$.
The marks in Salma's last $5$ English exams are: $6$, $7$, $8$, $5$ and $9$. a) Calculate the mean mark. b) Calculate the deviation of each mark from the mean (in…
The marks in Salma's last $5$ English exams are: $6$, $7$, $8$, $5$ and $9$.
a) Calculate the mean mark.
b) Calculate the deviation of each mark from the mean (in absolute value) and 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 these data…
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 these data in a frequency table and calculate:
- a) The range of the distribution.
- b) The mean deviation ($DM$) from the arithmetic mean.
A study was carried out on the battery life (in hours) of two different smartphone models, $A$ and $B$, conducting 10 tests with each. Model $A$:…
A study was carried out on the battery life (in hours) of two different smartphone models, $A$ and $B$, conducting 10 tests with each.
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 notice?
- b) Calculate the range and mean deviation of each model.
- c) If you had to buy one based on battery life reliability, which one would you choose and why?
Variance and standard deviation
Another way to keep the deviations from canceling out is to square them.
The goals scored by a futsal team in their last $5$ matches were: $2$, $5$, $1$, $4$, and $3$. a) Calculate the mean number of goals per match. b) Calculate the variance…
The goals scored by a futsal team in their last $5$ matches were: $2$, $5$, $1$, $4$, and $3$.
a) Calculate the mean number of goals per match.
b) Calculate the variance and standard deviation, rounding to two decimal places.
To study the dispersion of his marks ($5$, $6$, $7$, and $8$), a student subtracts the mean from each value and gets: $-1.5$; $-0.5$; $0.5$ and $1.5$. Since the sum of…
To study the dispersion of his marks ($5$, $6$, $7$, and $8$), a student subtracts the mean from each value and gets: $-1.5$; $-0.5$; $0.5$ and $1.5$. Since the sum of those differences is $0$, he claims: "The standard deviation is $0$, so my marks do not spread at all."
a) Explain the mistake he made in his reasoning.
b) Correctly calculate the standard deviation of the four marks, rounding to two decimal places.
The number of books read this term by the $20$ students in a class is given by this table: | Books read | $0$ | $1$ | $2$ | $3$ | |---|---|---|---|---| | No. of students…
The number of books read this term by the $20$ students in a class is given by this table:
| Books read | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|
| No. of students | $4$ | $8$ | $5$ | $3$ |
a) Calculate the mean number of books read.
b) Calculate the variance and standard deviation, rounding to two decimal places.
Comparing groups: center and spread
To decide between two groups, use two things: a measure of center (mean or median) and a measure of spread (range, $MD$, or $\sigma$).
Back to the beginning: the two cell phones with a 15 h mean are not equally reliable. The one with the smaller range, smaller $MD$, or smaller $\sigma$ is the one that lasts almost the same every time.
Two athletes train for the $100$ meters sprint and record their times with a smartwatch. Their last $5$ times, in seconds, are: Ana: $13.2$; $13.0$; $13.1$; $13.2$;…
Two athletes train for the $100$ meters sprint and record their times with a smartwatch. Their last $5$ times, in seconds, are:
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Ana: $13.2$; $13.0$; $13.1$; $13.2$; $13.0$
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Bea: $12.6$; $13.8$; $12.9$; $13.4$; $12.8$
a) Calculate the mean and range of times for each athlete.
b) Calculate the standard deviation for each athlete, rounding to two decimal places.
c) The coach must choose one of the two for the final and is looking for consistency. Who would you choose? Justify your answer.
Two basketball players, Lucía and María, scored the following points in their last 6 matches: Lucía: $18, 22, 20, 19, 21, 20$ María: $10, 35, 5, 40, 12, 18$ a) Calculate…
Two basketball players, Lucía and María, scored the following points in their last 6 matches:
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Lucía: $18, 22, 20, 19, 21, 20$
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María: $10, 35, 5, 40, 12, 18$
- a) Calculate the arithmetic mean of points for each player.
- b) Calculate the range and mean deviation for both players.
- c) If the coach is looking for a reliable player who ensures a consistent number of points, even if it is not very high, who should he choose? Justify your answer by comparing their dispersion.
Carlos compares two transportation apps to go to school. On his last $8$ trips, the waiting times in minutes were: App A: $4$, $6$, $5$, $7$, $5$, $6$, $4$, $7$ App B:…
Carlos compares two transportation apps to go to school. On his last $8$ trips, the waiting times in minutes were:
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App A: $4$, $6$, $5$, $7$, $5$, $6$, $4$, $7$
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App B: $2$, $9$, $3$, $10$, $4$, $8$, $3$, $5$
a) Calculate the mean waiting time for each app.
b) Calculate the standard deviation of each, rounding to two decimal places.
c) If Carlos has exactly $10$ minutes available each morning, which app would you recommend to him? Justify your answer using the mean and dispersion.
Measures of position: quartiles, deciles, and quintiles
The median divides the data into two halves. Other measures divide them into more equal parts.
For quartiles: $Q_2$ is the median, $Q_1$ is the median of the lower half, and $Q_3$ is the median of the upper half (if $N$ is odd, the median is not included in either half). For a decile $D_k$, calculate the position $\dfrac{k \cdot N}{10}$. If it is not a whole number, round up to the next position.
The box plot summarizes five values: minimum, $Q_1$, $Me$, $Q_3$, and maximum. Each section contains approximately 25% of the data.
A wide box means the data are more spread out in the middle. If a team's maximum in a box plot is less than some limit, you know it never exceeds it.
The marks of $11$ students in an exam were: $5$, $7$, $3$, $8$, $6$, $9$, $4$, $7$, $5$, $6$, and $10$. a) Order the marks and calculate the median. b) Calculate the…
The marks of $11$ students in an exam were: $5$, $7$, $3$, $8$, $6$, $9$, $4$, $7$, $5$, $6$, and $10$.
a) Order the marks and calculate the median.
b) Calculate the first and third quartiles.
c) Between which values is approximately the central $50\%$ of the marks concentrated?
Below is a box plot representing the marks obtained by the students of a class in a resit exam: Observe the graph and identify the following measures of position: a)…
Below is a box plot representing the marks obtained by the students of a class in a resit 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 obtained a mark higher than $8$?
In an esports competition, two teams, "Team Alpha" and "Team Beta", obtained the following response times (in milliseconds) in a reflexes test. The data is presented in…
In an esports competition, two teams, "Team Alpha" and "Team Beta", obtained the following response times (in milliseconds) in a reflexes 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 depends on never exceeding $200$ ms, which team has better prospects?