Mean, Median, and Mode
A small sneaker startup brags: "The average pay here is $\$3{,}000$ a month!" But six of the seven employees earn $\$2{,}000$ a month, and the owner takes $\$9{,}000$. Is $\$3{,}000$ a "typical" paycheck? The mean, the median, and the mode don't always tell the same story, and by the end of this page you'll know which one to trust.
The Three Measures of Center
A single number that describes the "middle" or "typical" value of a data set is a measure of center. There are three common ones.
Why the median uses the middle: half of the data lies at or below it and half at or above it. If $N$ is odd, the median sits at position $\frac{N+1}{2}$ of the ordered list.
Change the data below and watch how the three measures react.
If $N$ is odd it's even easier. For $3, 5, 5, 6, 8$ the middle value is $5$, so the median is $5$.
In a botany experiment, the number of leaves of $8$ plants of the same species has been measured after two weeks of growth. The results obtained are:…
In a botany experiment, the number of leaves of $8$ plants of the same species has been measured after two weeks of growth. The results obtained are: $12, 15, 14, 12, 16, 12, 18, 13$.
Calculate the arithmetic mean, the median and the mode of this data.
Given the following two data sets referring to the number of books read in a year by two groups of young people: Set A: $2, 4, 5, 7, 9$ Set B: $1, 3, 4, 6, 8, 10$ Find…
Given the following two data sets referring to the number of books read in a year by two groups of young people:
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Set A: $2, 4, 5, 7, 9$
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Set B: $1, 3, 4, 6, 8, 10$
Find the median for each of the sets and explain the difference in the calculation procedure when having an even or odd number of data points.
Frequency Tables and Bar Charts
When a value repeats many times, it is shorter to write how often it occurs. The frequency $f$ of a value is the number of times it appears. A bar chart shows the same information: the height of each bar is the frequency.
Here is the number of pets owned by $25$ students:
| Pets $x$ | Frequency $f$ | $x \cdot f$ |
|---|---|---|
| 0 | 5 | 0 |
| 1 | 9 | 9 |
| 2 | 6 | 12 |
| 3 | 3 | 9 |
| 4 | 2 | 8 |
| Total | $N = 25$ | $38$ |
Categorical data. If the data are categories (colors, pizza toppings, brands), you can't add or order them. Only the mode makes sense. If $9$ students pick pepperoni, $7$ cheese, and $4$ veggie, the mode is pepperoni. A "mean topping" has no meaning.
The ages of the participants in a robotics workshop are summarized in the following frequency table: $14$ years old: $5$ people $15$ years old: $8$ people $16$ years…
The ages of the participants in a robotics workshop are summarized in the following frequency table:
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$14$ years old: $5$ people
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$15$ years old: $8$ people
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$16$ years old: $4$ people
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$17$ years old: $3$ people
Determine the mean age, the median age and the mode.
The following bar chart shows the number of siblings the students in a class have. From the chart, construct the frequency table and calculate the mode and the…
The following bar chart shows the number of siblings the students in a class have.
From the chart, construct the frequency table and calculate the mode and the arithmetic mean.
$20$ people were asked for their favorite color for the interface design of a new mobile app. The results are: Blue: $8$ people Green: $5$ people Red: $4$ people Yellow:…
$20$ people were asked for their favorite color for the interface design of a new mobile app. The results are:
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Blue: $8$ people
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Green: $5$ people
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Red: $4$ people
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Yellow: $3$ people
State which is the mode. Does it make sense to calculate the mean or the median in this case? Explain your answer.
Outliers: When the Mean Misleads
The mean uses every data value, so one extreme value, called an outlier, drags it toward itself. The median only depends on order, so it barely moves.
Now back to the startup. Here is the pay data on a number line.
In a quality control test, the time in seconds it takes for a computer to boot up is measured. The following times are obtained: $12, 11, 13, 12, 14, 85$. Calculate the…
In a quality control test, the time in seconds it takes for a computer to boot up is measured. The following times are obtained: $12, 11, 13, 12, 14, 85$.
Calculate the mean and the median. Which of the two measures better represents the central tendency of this data? Justify your answer.
In a small technology company, the monthly salaries of its $7$ employees are: $1200 \text{ \$}, 1250 \text{ \$}, 1300 \text{ \$}, 1200 \text{ \$}, 1400 \text{ \$}, 1350…
In a small technology company, the monthly salaries of its $7$ employees are:
$1200 \text{ \$}, 1250 \text{ \$}, 1300 \text{ \$}, 1200 \text{ \$}, 1400 \text{ \$}, 1350 \text{ \$}, 8500 \text{ \$}$.
Calculate the mean and the median. Write a short text explaining to future employees which of the two measures more realistically represents the usual salary they will earn and why.
Back to the start: $\$3{,}000$ is not a typical paycheck. Six of the seven employees earn $\$2{,}000$. The owner's $\$9{,}000$ is an outlier that pulls the mean up, while the median, $\$2{,}000$, describes what a new employee will usually earn.
Working Backward from the Mean
The mean formula can be turned around. Multiply both sides by $N$:
This turns "find the missing value" problems into one-step equations. Here the five scores below are drawn as gaps from their mean of $85$: the gaps above and below the mean cancel out.
The same idea finds an unknown frequency. Suppose people at a party ate $1$ slice ($4$ people), $2$ slices ($6$ people), $3$ slices ($f$ people), or $4$ slices ($5$ people), and the mean was $2.5$ slices.
A 10th-grade student has taken five math exams throughout the term. His grades on the first four are: $6, 7, 5$ and $8$. If he wants his exact average grade to be a $7$,…
A 10th-grade student has taken five math exams throughout the term. His grades on the first four are: $6, 7, 5$ and $8$.
If he wants his exact average grade to be a $7$, what grade must he get on the fifth exam?
When calculating the mean of the grades of $25$ students, the teacher obtained a $6,4$. Later, reviewing the exams, he realized that he had mistakenly written down a $3$…
When calculating the mean of the grades of $25$ students, the teacher obtained a $6,4$. Later, reviewing the exams, he realized that he had mistakenly written down a $3$ for a student whose actual grade was a $8$.
Calculate the new correct mean without needing to know all the individual grades.
In a shoe store, the number of pairs sold according to size during a weekend has been recorded. The mean size sold was $39,5$. Knowing the following data, determine the…
In a shoe store, the number of pairs sold according to size during a weekend has been recorded. The mean size sold was $39,5$. Knowing the following data, determine the number of pairs sold of size $40$ (represented by $f$).
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Size $38$: $4$ pairs
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Size $39$: $8$ pairs
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Size $40$: $f$ pairs
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Size $41$: $5$ pairs
Weighted Mean
Not every value counts the same. In many classes, a final exam is worth more than homework. Then each value is multiplied by its weight before adding.
Weighted means also answer "what do I need?" questions, using an inequality.
The final grade for the Physics course is calculated using a weighted mean. The lab assignments are worth $20\%$, the research projects $30\%$ and the final exam $50\%$.…
The final grade for the Physics course is calculated using a weighted mean. The lab assignments are worth $20\%$, the research projects $30\%$ and the final exam $50\%$.
If a student has a $8$ on the labs, a $6$ on the projects and a $7,5$ on the final exam, calculate her final average grade.
In the assessment of a subject, the final grade is calculated using three components: Mid-term exam (weighting $30\%$) Research project (weighting $20\%$) Final exam…
In the assessment of a subject, the final grade is calculated using three components:
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Mid-term exam (weighting $30\%$)
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Research project (weighting $20\%$)
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Final exam (weighting $50\%$)
A student has a $4,5$ in the mid-term exam and a $6$ in the project. The final exam grade must be an integer or have exactly one decimal place. What is the minimum grade they must get in the final exam for their weighted mean to be at least $5,0$? If the maximum grade is $10$, is it possible for them to achieve a mean of $8,0$?
Combining Groups and Mixtures
You can also use a weighted mean when the weights are the sizes of groups. You can't just average two group means. A bigger group should count more.
When one of the weights is the unknown, write an equation.
The same idea works for mixtures. If a mix is made of a fraction $x$ of one ingredient and $1 - x$ of another, the mean value of the mix is a weighted mean with those fractions as weights.
A logistics company has two industrial warehouses. In the Dallas warehouse, $45$ workers work with a mean salary of $1800 \text{ \$}$. In the Chicago warehouse, $30…
A logistics company has two industrial warehouses. In the Dallas warehouse, $45$ workers work with a mean salary of $1800 \text{ \$}$. In the Chicago warehouse, $30$ workers work with a mean salary of $2100 \text{ \$}$.
Calculate the mean salary of all the workers in the company as a whole.
An investor buys shares in a company for three consecutive months. The first month he buys $n$ shares at a mean price of $12 \text{ \$}…
An investor buys shares in a company for three consecutive months. The first month he buys $n$ shares at a mean price of $12 \text{ \$}$. The second month he buys twice as many shares as the first month at a mean price of $15 \text{ \$}$. The third month he buys $50$ shares at $10 \text{ \$}$.
If the overall mean price of all the shares bought in the three months is $13 \text{ \$}$, how many shares did he/she buy in the first month?
A metal alloy is composed of two metals $A$ and $B$. The density of metal $A$ is $3 \text{ g/cm}^3$ and that of metal $B$ is $8 \text{ g/cm}^3$. If the mean density of…
A metal alloy is composed of two metals $A$ and $B$. The density of metal $A$ is $3 \text{ g/cm}^3$ and that of metal $B$ is $8 \text{ g/cm}^3$.
If the mean density of the resulting alloy is $6 \text{ g/cm}^3$, what percentage of the total volume of the alloy corresponds to metal $B$? (Note that the mean density of an alloy behaves like the weighted mean of the densities of its components with respect to their volume fractions).
Grouped Data
Sometimes data come in intervals (classes), as in a histogram, and the individual values are lost. We pretend every value in a class equals the class's midpoint. The interval $[a, b)$ includes $a$ but not $b$, and its midpoint is $\frac{a + b}{2}$.
Here are the daily minutes of streaming for $30$ students.
When two frequencies are missing, use two equations: one for the total count and one for the mean.
In a study on water consumption in a community, the monthly usage in cubic feet ($m^3$) of several homes has been recorded, grouped into the following intervals:…
In a study on water consumption in a community, the monthly usage in cubic feet ($m^3$) of several homes has been recorded, grouped into the following intervals:
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$[0, 10)$: $12$ homes
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$[10, 20)$: $25$ homes
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$[20, 30)$: $18$ homes
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$[30, 40)$: $5$ homes
Calculate the mean water consumption and state which is the modal class.
The following histogram represents the time (in minutes) that a group of people spend playing sports per day. Calculate the mean time spent playing sports from the class…
The following histogram represents the time (in minutes) that a group of people spend playing sports per day.
Calculate the mean time spent playing sports from the class midpoints extracted from the chart.
In a medical center, the cholesterol level of $50$ patients has been recorded. The arithmetic mean of the distribution is $194 \text{ mg/dL}$. In the table, two…
In a medical center, the cholesterol level of $50$ patients has been recorded. The arithmetic mean of the distribution is $194 \text{ mg/dL}$. In the table, two frequencies, $f_1$ and $f_2$, have been accidentally erased.
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$[160, 180)$: $10$ patients
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$[180, 200)$: $f_1$ patients
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$[200, 220)$: $f_2$ patients
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$[220, 240)$: $8$ patients
Set up and solve a system of equations to find the absolute frequencies $f_1$ and $f_2$.
Median of Grouped Data and the Cumulative Graph
For grouped data you can't pick out the middle value, but you can find the class that contains it and then estimate. Make a column of cumulative frequencies (running totals) and find the first class whose total reaches $\frac{N}{2}$.
A cumulative frequency graph (also called an ogive) plots the running totals above the class boundaries and joins the points with segments. To find the median, start at height $\frac{N}{2}$ on the vertical axis, go across to the graph, and then down to the horizontal axis. This is exactly the interpolation above.
The slope of each segment is $\frac{f}{w}$, so the steepest segment belongs to the class with the most values. That is the modal class (when the classes have equal widths). Here the steepest segment is $[20, 30)$, which has the most workers.
The weight in pounds of a group of $40$ packages in a post office has been measured, obtaining the following distribution: $[0, 5)$: $6$ packages $[5, 10)$: $14$…
The weight in pounds of a group of $40$ packages in a post office has been measured, obtaining the following distribution:
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$[0, 5)$: $6$ packages
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$[5, 10)$: $14$ packages
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$[10, 15)$: $12$ packages
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$[15, 20)$: $8$ packages
Find the interval where the median is located and estimate its exact value using the linear interpolation formula.
The following cumulative frequency polygon (ogive) shows the distribution of the distances obtained by a group of $100$ athletes in the long jump event. From the graph,…
The following cumulative frequency polygon (ogive) shows the distribution of the distances obtained by a group of $100$ athletes in the long jump event.
From the graph, estimate the value of the median. If the data were grouped in a frequency table with the intervals of the horizontal axis, what would be the modal class? Justify your answers based on the gradients of the graph.
What Happens to the Mean When the Data Change?
Suppose a teacher adds $5$ points to every score, or a store raises all prices by $10\%$. How do the center measures change?
In words: adding $b$ to everything shifts the center by $b$. Multiplying everything by $a$ multiplies the center by $a$.
The mean height of a basketball team of $10$ players is $192 \text{ cm}$. If all the players decide to wear a special model of sneakers that increases their height by…
The mean height of a basketball team of $10$ players is $192 \text{ cm}$.
If all the players decide to wear a special model of sneakers that increases their height by $4 \text{ cm}$, what will be the new mean height of the team? Will the mode change if there was a more frequent height?
Let $X$ be a statistical variable with mean $\bar{x} = 15$. A new variable $Y$ is defined by the linear transformation $y_i = 3x_i - 4$. Prove algebraically, applying…
Let $X$ be a statistical variable with mean $\bar{x} = 15$. A new variable $Y$ is defined by the linear transformation $y_i = 3x_i - 4$.
Prove algebraically, applying the properties of summation and the arithmetic mean, that the mean of the new variable is $\bar{y} = 3\bar{x} - 4$, and calculate its numerical value.
Putting It All Together: Reasoning Problems
The hardest problems give you clues about the mean, median, mode, and range, or tell you how adding a value changes the mean. The plan is always the same: turn every clue into an equation or an inequality.
Clue → math
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Mean $= m$ for $N$ values: the sum is $m \cdot N$.
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Median of five ordered values $a \le b \le c \le d \le e$: it is $c$.
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Range: $e - a$.
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Adding a new value $D$: new sum $=$ old sum $+\, D$ and new count $= N + 1$.
The mean of a set of $n$ data values is $\bar{x}$. If we add a new data value of $D$, the new mean increases by $2$ units. If instead of $D$, we add a data value of…
The mean of a set of $n$ data values is $\bar{x}$. If we add a new data value of $D$, the new mean increases by $2$ units. If instead of $D$, we add a data value of $D - 15$, the new mean decreases by $1$ unit with respect to the original mean $\bar{x}$.
Set up the corresponding equations and calculate the original number of data values $n$.
There is a set of five positive integers ordered from smallest to largest: $a, b, c, d, e$. It is known that they simultaneously satisfy the following conditions: The…
There is a set of five positive integers ordered from smallest to largest: $a, b, c, d, e$.
It is known that they simultaneously satisfy the following conditions:
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The mode is $4$.
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The median is $5$.
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The arithmetic mean is $6$.
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The range (difference between the largest and the smallest) is $7$.
Determine the exact value of the five numbers, reasoning your answer step by step.