Measures of Center and Spread
Jayden and Maya both average 20 points per game on the school basketball team. Jayden scores about 20 almost every night. Maya alternates between huge games and duds. If you were the coach, who would you pick for the last shot of a championship game? One number can't tell you. You need to know where the center of the data is and how spread out it is.
Mean, median, and mode
Statisticians study a population, the whole group they want to learn about (for example, all the teenagers in a study of TikTok use). The numbers they collect are the data. A number that summarizes the data is called a statistic or parameter. The first three summaries describe the center.
Change the data and watch where the mean and the median land:
Why divide by $N$? The mean is what each person would have if the total were shared equally. Why the median in the middle? Because it splits the ordered list into two halves with the same number of values.
In a study on the use of social media, a group of teenagers were asked how many hours a day they spend watching videos on TikTok. The results obtained are:…
In a study on the use of social media, a group of teenagers were asked how many hours a day they spend watching videos on TikTok. The results obtained are:
$2, 3, 1, 4, 2, 3, 2, 5, 2, 1$
Identify the population of the study and calculate the mode of the hours of use.
Seven friends compare the number of unread messages they have on their messaging app. The data is as follows: $12, 15, 8, 20, 15, 14, 11$ Calculate the arithmetic mean…
Seven friends compare the number of unread messages they have on their messaging app. The data is as follows:
$12, 15, 8, 20, 15, 14, 11$
Calculate the arithmetic mean ($\bar{x}$) of the number of pending messages for this group.
During a week of training, an athlete has recorded the miles she has run each day to prepare for a race: $5, 8, 10, 12, 15, 18, 21$ Find the median ($Me$) of the…
During a week of training, an athlete has recorded the miles she has run each day to prepare for a race:
$5, 8, 10, 12, 15, 18, 21$
Find the median ($Me$) of the distance run. What is the meaning of this value in relation to the total number of days trained?
Reading data from graphs
Often the data don't come as a list. A dot plot has one dot per data value, stacked above a number line. A line graph shows one value per point (for example, the high temperature each day). First read every value off the graph and write them as a list. Then compute as usual.
The following graph shows the maximum temperatures recorded in a mountain lodge during the first 5 days of January. Knowing that the temperatures have been…
The following graph shows the maximum temperatures recorded in a mountain lodge during the first 5 days of January.
Knowing that the temperatures have been $4^\circ C, 6^\circ C, 2^\circ C, 8^\circ C$ and $5^\circ C$, calculate the range of the temperature over those days.
Look at the following representation of a data set on a number line. The dots represent the number of pets that several neighbors have: From the graph, write the list of…
Look at the following representation of a data set on a number line. The dots represent the number of pets that several neighbors have:
From the graph, write the list of data and calculate the mode ($Mo$) and the range.
Which measure of center is best?
The three measures can tell different stories, especially when one value is far from all the others. Such a value is an outlier.
A group of 20 teenagers has recorded how many times they have been to the movies in the last six months. The data obtained is as follows:…
A group of 20 teenagers has recorded how many times they have been to the movies in the last six months. The data obtained is as follows:
$2, 3, 0, 1, 5, 3, 2, 1, 0, 0, 2, 4, 3, 2, 1, 1, 6, 2, 0, 3$
Organize this data into a frequency table and calculate the measures of central tendency: the arithmetic mean ($\bar{x}$), the median ($Me$) and the mode ($Mo$). Which of them do you consider best represents the group?
A youth clothing store wants to analyze the age of its customers. To do this, it takes a sample of 10 people on a Saturday morning:…
A youth clothing store wants to analyze the age of its customers. To do this, it takes a sample of 10 people on a Saturday morning:
$14, 15, 16, 14, 22, 15, 17, 14, 35, 16$
- a) Calculate the range and the mean age.
- b) The median is $15,5$ years. If we remove the 35-year-old customer (who is considered an outlier), calculate the new mean and the new median. Which parameter has changed the most? Why do you think this happens?
In a small company, the salaries of the $5$ employees are: $\$1100, \$1200, \$1250, \$1300$ and $\$1400$. The manager has a salary of $\$4500$. a) Calculate the mean and…
In a small company, the salaries of the $5$ employees are: $\$1100, \$1200, \$1250, \$1300$ and $\$1400$. The manager has a salary of $\$4500$.
- a) Calculate the mean and the median of the salaries including the manager.
- b) Which of the two measures better represents the "typical" salary in the company? Explain the effect the manager's salary (extreme value) has on the mean and why the median is a more robust parameter in this case.
- c) If they want to give an image of prosperity in the company's advertising, which parameter would be used? Would it be misleading?
Working with totals
The mean hides a useful fact: $\text{mean} \times N = \text{sum of the data}$. When a value is added, removed, or unknown, work with the total, not the mean.
Building a data set from clues
Some problems give you the mean, median, mode, or range and ask you to invent data. Place the clues one at a time.
With 6 values, the median is the mean of the 3rd and 4th values. Plan those two first.
Sarah's grades in her first four math exams were $6,5$, $4$, $8$ and $7,5$. If she still has a fifth exam to take, what grade must she get for her final mean to be…
Sarah's grades in her first four math exams were $6,5$, $4$, $8$ and $7,5$. If she still has a fifth exam to take, what grade must she get for her final mean to be exactly $7$? Justify your answer by setting up an equation.
In a group of $10$ friends, their mean height is $172$ cm. If a person who is $194$ cm tall joins the group, what will the new mean be? If another person who was $161$…
In a group of $10$ friends, their mean height is $172$ cm. If a person who is $194$ cm tall joins the group, what will the new mean be? If another person who was $161$ cm tall then leaves, what will the final mean of the group be? Justify step by step how each change affects the total sum of the heights to carry out the calculation.
Make up a series of $6$ statistical data points (positive integers) that simultaneously meet the following conditions: 1. The mode is $4$. 2. The median is $5$. 3. The…
Make up a series of $6$ statistical data points (positive integers) that simultaneously meet the following conditions:
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The mode is $4$.
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The median is $5$.
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The range is $10$.
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The mean is greater than the median.
Show the calculations that verify your series meets all the requirements.
Quartiles and box plots
The median splits ordered data into two halves. Quartiles split it into four parts with about 25% of the data each.
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$Q_1$ (first quartile): the median of the lower half. About 25% of the data are at or below it.
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$Q_2$: the median of all the data (50%).
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$Q_3$ (third quartile): the median of the upper half. About 75% of the data are at or below it.
The five-number summary is minimum, $Q_1$, median, $Q_3$, maximum. A box plot (box-and-whisker plot) draws it: the box goes from $Q_1$ to $Q_3$ with a line at the median, and the whiskers reach the minimum and the maximum.
The box holds the middle 50% of the data. The distance $Q_3 - Q_1$ is the interquartile range (IQR). It measures how spread out the middle half is.
When $N$ is odd, leave the median out of both halves.
Reading a box plot
Each part of the plot (whisker, half of the box) contains about 25% of the data. Suppose a survey of reading time has $\text{min} = 0$, $Q_1 = 2$, median $= 4$, $Q_3 = 7$, $\text{max} = 12$. A student who reads 6 hours is between the median (4) and $Q_3$ (7), so they are in the 50%–75% part. About 75% of students read 7 hours or less.
The height of a group of 8th grade students has been measured and the data has been ordered from lowest to highest. The values of the quartiles obtained are:…
The height of a group of 8th grade students has been measured and the data has been ordered from lowest to highest. The values of the quartiles obtained are:
$Q_1 = 158 \text{ cm}, Q_2 = 165 \text{ cm}, Q_3 = 172 \text{ cm}$
- a) What is the value of the median?
- b) What percentage of students measure $172 \text{ cm}$ or less?
A survey has been carried out on the number of hours per week that 8th grade students spend reading books for pleasure. The measures of position obtained are: $Mín = 0$;…
A survey has been carried out on the number of hours per week that 8th grade students spend reading books for pleasure. The measures of position obtained are:
$Mín = 0$; $Q_1 = 2$; $Me = 4$; $Q_3 = 7$; $Máx = 12$
Draw the corresponding box-and-whisker plot (box plot). If a student reads 6 hours a week, in which quarter of the population are they (between 0-25%, 25-50%, 50-75% or 75-100%)?
Look at the following box plot representing the weekly hours of use of a mobile app in a sample of teenagers: Based on the graph, answer: a) What is the value of the…
Look at the following box plot representing the weekly hours of use of a mobile app in a sample of teenagers:
Based on the graph, answer:
- a) What is the value of the median, the lower quartile ($Q_1$) and the upper quartile ($Q_3$)?
- b) Calculate the interquartile range ($RIC$).
- c) Reason whether the distribution is symmetrical or if it shows skewness. In which half of the data is there greater spread?
Spread: range and standard deviation
The measures of spread tell how far the data are from each other or from the mean.
Some problems write the variance as $S^2$. It is the same idea. Why square the distances? Values below the mean give negative distances and values above give positive ones, and they cancel out (their sum is always 0). Squaring makes every distance positive.
The range is quick but only uses two values. The standard deviation uses every value. When two data sets have the same mean, the one with the smaller standard deviation is more consistent: its values cluster near the mean.
Two e-sports teams have recorded their scores in the last 5 matches. Team A: $85, 88, 90, 92, 95$ Team B: $60, 75, 90, 105, 120$ Both teams have the same mean,…
Two e-sports teams have recorded their scores in the last 5 matches.
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Team A: $85, 88, 90, 92, 95$
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Team B: $60, 75, 90, 105, 120$
Both teams have the same mean, $\bar{x} = 90$. Without performing complex standard deviation calculations, look at the data and explain which of the two teams is more consistent (has less dispersion) and why.
In a Physical Education class, the recovery time of two groups after a race has been measured. Both groups have the same mean heart rate, $\bar{x} = 85$ bpm. However,…
In a Physical Education class, the recovery time of two groups after a race has been measured. Both groups have the same mean heart rate, $\bar{x} = 85$ bpm. However, the variance of Group 1 is $S^2 = 16$ and that of Group 2 is $S^2 = 100$.
- a) Calculate the standard deviation of each group.
- b) In which of the two groups do the students have a more similar fitness level to each other? Justify your answer based on the dispersion.
Look at the following two sets of data representing the points scored by two basketball players in their last 5 matches: Player A: $18, 20, 22, 19, 21$ Player B:…
Look at the following two sets of data representing the points scored by two basketball players in their last 5 matches:
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Player A: $18, 20, 22, 19, 21$
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Player B: $10, 30, 5, 25, 30$
Without performing complex variance calculations, look at the data and answer:
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Do they have the same arithmetic mean?
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Which of the two players is more "consistent" in their scores?
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If you had to choose one to play the last minute of a final, which one would you choose and why? Use the concepts of mean and dispersion in your answer.
Frequency tables and grouped data
When values repeat, a frequency table lists each value $x$ and how many times it occurs, the frequency $f$. Then $N$ is the sum of the frequencies, not the number of rows.
For data in intervals like $[0, 10)$, we no longer know the exact values. Use the class mark $x$, the midpoint of each interval: $\dfrac{\text{lower} + \text{upper}}{2}$. Then use the same formulas.
Grouping loses information, so values computed from class marks are estimates. Suppose the data in $[0, 10)$ and $[10, 20)$ were really $1, 9, 11, 19$. The class marks $5, 5, 15, 15$ give $\sigma = 5$, but the real standard deviation is $\sqrt{41} \approx 6.4$.
A teacher has organized the number of spelling mistakes their students made in the last dictation into a frequency table: | Mistakes ($x_i$) | No. of students ($f_i$) |…
A teacher has organized the number of spelling mistakes their students made in the last dictation into a frequency table:
| Mistakes ($x_i$) | No. of students ($f_i$) |
|---|---|
| $0$ | $8$ |
| $1$ | $12$ |
| $2$ | $6$ |
| $3$ | $4$ |
Calculate the mean number of mistakes in the class using the data from the table.
An e-sports club has recorded the time (in minutes) its members take to complete a difficult level of a video game. The results are shown in this grouped data table: |…
An e-sports club has recorded the time (in minutes) its members take to complete a difficult level of a video game. The results are shown in this grouped data table:
| Time (min) | [10, 20) | [20, 30) | [30, 40) | [40, 50) |
|---|---|---|---|---|
| No. of players | 4 | 12 | 8 | 6 |
Calculate the arithmetic mean and the standard deviation of the distribution using the midpoints.
A courier company analyzes the weight of the packages it transports. The data has been grouped in the following interval frequency table: | Weight in lbs $[L_i, L_s)$ |…
A courier company analyzes the weight of the packages it transports. The data has been grouped in the following interval frequency table:
| Weight in lbs $[L_i, L_s)$ | No. of packages ($f_i$) |
|---|---|
| $[0, 2)$ | $12$ |
| $[2, 4)$ | $25$ |
| $[4, 6)$ | $38$ |
| $[6, 8)$ | $15$ |
| $[8, 10)$ | $10$ |
Calculate the arithmetic mean, the variance and the standard deviation of the distribution. To do this, first identify the class marks ($x_i$) for each interval.
Comparing spread with the coefficient of variation
Is a standard deviation of 5 big or small? It depends on the mean. A spread of 5 minutes matters when the mean is 10 minutes, and hardly matters when the mean is 500 minutes. The coefficient of variation compares spread relative to the mean.
Back to the start: Jayden has $\sigma \approx 1.4$ and $CV \approx 1.4 \div 20 \approx 7\%$. Maya has $\sigma \approx 11.2$ and $CV \approx 56\%$. For a coach who needs a reliable 20 points, Jayden is the safer pick. If the team needs a huge game to win and a dud costs nothing extra, Maya's spread is a gamble worth taking.
Two messaging applications, "App A" and "App B", are competing to be the most stable. The response time (in milliseconds) has been measured in 10 different attempts,…
Two messaging applications, "App A" and "App B", are competing to be the most stable. The response time (in milliseconds) has been measured in 10 different attempts, obtaining the following results:
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App A: $\bar{x} = 120$ ms; $\sigma = 15$ ms
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App B: $\bar{x} = 125$ ms; $\sigma = 5$ ms
Calculate the coefficient of variation ($CV$) for both applications and explain with reasons which of the two is more "reliable" or consistent in its operation for a user.
Two eSports players, Alex and Bea, have recorded their scores in the last $8$ matches of a tournament. Alex: $85, 90, 88, 92, 87, 89, 91, 88$ Bea:…
Two eSports players, Alex and Bea, have recorded their scores in the last $8$ matches of a tournament.
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Alex: $85, 90, 88, 92, 87, 89, 91, 88$
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Bea: $70, 98, 65, 100, 75, 95, 80, 100$
- a) Calculate the mean and standard deviation for both players.
- b) Use the coefficient of variation ($CV$) to determine which of the two is a more consistent and reliable player for the team. Explain what it means in this context for one player to have a much higher standard deviation than the other if their means were similar.
Changing the data and the bell curve
What happens to the mean and standard deviation when every value changes in the same way?
A 10% increase multiplies by $k = 1.10$.
The 68–95–99.7 rule
Many data sets (heights, test scores, machine parts) form a symmetric bell curve. For these, about 68% of the data are within $1\sigma$ of the mean, 95% are within $2\sigma$, and 99.7% are within $3\sigma$.
To see how far a value is from the mean in standard deviations, compute $\dfrac{x - \bar{x}}{\sigma}$. A value more than $3\sigma$ from the mean is very rare (about 0.3% of the data), so it counts as an outlier. A value 2σ away is unusual but not extreme.
A teacher has graded the exams of a class of $25$ students. The mean was $6,4$ and the standard deviation was $1,2$. After reviewing one of the exercises, they decide to…
A teacher has graded the exams of a class of $25$ students. The mean was $6,4$ and the standard deviation was $1,2$. After reviewing one of the exercises, they decide to add $0,5$ points to all the students in the class.
- a) What will the new mean and the new standard deviation be?
- b) If instead of adding a fixed amount, they decided to increase all the grades by $10\%$, how would both parameters be affected? Justify your answer based on the properties of statistical parameters.
It is known that in a very large data distribution that follows a "bell curve", $68\%$ of the data is found in the interval $(\bar{x} - \sigma, \bar{x} + \sigma)$. In a…
It is known that in a very large data distribution that follows a "bell curve", $68\%$ of the data is found in the interval $(\bar{x} - \sigma, \bar{x} + \sigma)$.
In a light bulb factory, the mean lifespan of a model is $1200$ hours with a standard deviation of $150$ hours. If $5000$ light bulbs are produced:
- a) How many light bulbs are expected to last between $1050$ and $1350$ hours?
- b) If a light bulb lasts more than $1500$ hours, how many standard deviations from the mean is it? Do you consider it to be a common value or an outlier?