One-variable data: distributions and measures of center and spread

Each question is about one batch of numbers, given as a list, a table, a bar graph, a dot plot, a histogramA bar chart that shows how many values fall in each range. or a box plot. The last line usually reads like one of these:

• “What is the mean of the data shown?”
• “The frequency table summarizes the $57$ data values in a data set. What is the maximum data value in the data set?”
• “Which of the following correctly compares the medians and the ranges?”
• “Which of the following lists represents a data set with the smallest standard deviationHow spread out a set of numbers is around its average.?”

Others give the mean and ask for a missing value.

Here's what each word means, worked on the list $2, 3, 3, 4, 13$.

1. Mean: add every value.

   $$2+3+3+4+13=25$$

2. Then divide by how many values there are.

   $$25\div5=5$$

3. Median: the middle value once the list is in order.

   $$2,\ 3,\ \mathbf{3},\ 4,\ 13$$

4. Mode: the value that shows up most often.

   $$2,\ \mathbf{3},\ \mathbf{3},\ 4,\ 13$$

5. Range: the largest value minus the smallest.

   $$13-2=11$$

With an even count, the median is halfway between the two middle values. Find the median of $18, 44, 26, 40, 12, 30$.

1. Sort from least to greatest.

   $$12,\ 18,\ 26,\ 30,\ 40,\ 44$$

2. Six values, so average the 3rd and 4th.

   $$\frac{26+30}{2}=28$$

Each dot is one value. The numbers along the bottom are the values, and a stack's height is how many times that value appears.

1. Count the values: add the stack heights.

   $$2+3+3+5+2=15$$

2. With 15 values, the median is the 8th.

   $$\frac{15+1}{2}=8$$

3. Count up from the left to the 8th place. The median is 24.

   $$20,20,22,22,22,24,24,\mathbf{24}$$

4. Range: largest value minus smallest.

   $$28-20=8$$

5. Mode: the value under the tallest stack.

   $$26$$

A bar graph of frequencies reads the same way.

A frequency table lists each value once, with how many times it appears.

For a mean, each value counts as many times as it appears. Multiply each value by its frequency and add to get the total. Then divide by the total of the frequencies. A common wrong choice divides by the number of rows instead.

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Ilona recorded the bike shop’s tire shipments.

What is the mean number of tires per shipment?

Show the answer

Accepted answer: 78

Solution

Multiply each tire count by its number of shipments, then add: $44(2)+66(7)+77(6)+89(8)+113(2)=1{,}950$. The total number of shipments is $2+7+6+8+2=25$, so the mean is $\frac{1{,}950}{25}=78$.

A common wrong answer is $390$, which results from dividing $1{,}950$ by the 5 table rows instead of the 25 shipments. Another common wrong answer is $77$, which is the medianThe middle value once the numbers are put in order.Read the lesson of the expanded 25-value data set, not the mean.

Report a problem

For a median, count up through the frequencies. Take values 20, 25 and 30 with frequencies 7, 4 and 2.

1. Count the values.

   $$7+4+2=13$$

2. With 13 values, the median is the 7th.

   $$\frac{13+1}{2}=7$$

3. The 20s fill places 1 to 7, so the median is 20, even though 25 is the middle row.

   $$20$$

Some questions make a frequency the unknown. With frequencies 7, 4 and $n$, what's the smallest $n$ that makes the median 30?

1. The 30s start at place 12, so the middle has to be place 12 or later.

   $$7+4=11$$

2. The median averages places 11 and 12, a 25 and a 30. That's 27.5.

   $$n=11:\ 22 \text{ values}$$

3. The median is place 12, a 30. So the smallest $n$ is 12.

   $$n=12:\ 23 \text{ values}$$

Group A has 12 values with a mean of 70. Group B has 3 values with a mean of 90. What's the mean of all 15?

1. Group A's total: mean times count.

   $$12\times70=840$$

2. Group B's total.

   $$3\times90=270$$

3. Add the totals and divide by all 15 values.

   $$(840+270)\div15=74$$

The answer lands between the two means, closer to the mean of the group with more values.

Here $n_1$ and $n_2$ are the group sizes, and $m_1$ and $m_2$ are their means.

A histogram is a bar graph where each bar covers an interval of values, like 100 to 120. A bar's height is how many values fall in that interval.

1. Count the values: add the bar heights.

   $$3+2+3+12=20$$

2. With 20 values, the median averages these two.

   $$10\text{th and }11\text{th}$$

3. The first three bars hold only 8 values, so both middle places are in the last bar.

   $$3+2+3=8$$

A box plot shows five numbers: the minimum, the first quartileThe value a quarter of the way through sorted data, or three quarters of the way through. The box in a box plot runs between those two values. $Q_1$, the median, the third quartile $Q_3$, and the maximum. $Q_1$ is the middle of the lower half of the sorted data, and $Q_3$ is the middle of the upper half.

The interquartile rangeThe spread of the middle half of the data, found by subtracting the lower quartile from the upper quartile., or IQR, is the width of the box.

1. Range: one whisker end to the other.

   $$50-20=30$$

2. IQR: $Q_3-Q_1$.

   $$44-27=17$$

“There is not enough information” is right only when the display lacks what you need, like a mean from a box plot. It's never right when the full lists are printed.

Standard deviation measures how far the values usually sit from their mean. On the test you only compare them. These two lists both have a mean of 5 and a range of 10.

1. Distances from 5: 5, 0, 0, 5.

   $$0,\ 5,\ 5,\ 10$$

2. Distances from 5: 5, 5, 5, 5.

   $$0,\ 0,\ 10,\ 10$$

If a question prints one histogram and describes another in words, sketch the second beside the first. The one with more values in the middle bars has the smaller standard deviation.

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At Iker's quarry, the graph gives blast-hole spacings for lines A, B, and C. Line D's spacings are 14 and 22 feet.

Which line has the smallest standard deviation in spacing?

1. A

   Line A

2. B

   Line B

3. C

   Line C

4. D

   Line D

Show the answer

• A

  Line A

  This choice judges spread from the endpoint frequencies while overlooking the four spacings at both 16 and 20 feet. All four lines have the same range, and Line A has substantial frequency away from its mean.

• B

  Line B

  This choice uses Line B's smaller medianThe middle value once the numbers are put in order.Read the lesson as evidence of a smaller standard deviationHow spread out a set of numbers is around its average.Read the lesson. The median describes the center, so a smaller median does not mean the data are less spread out.

• C

  Line C

  Correct answer

  Line C is centered at 18 feet, and 12 of its 18 spacings equal 18. Its variance is $4$, which is smaller than the variances for lines A, B, and D, so it has the smallest standard deviationHow spread out a set of numbers is around its average.Read the lesson.

• D

  Line D

  This choice treats having only two observations as a guarantee of less spread. The observations 14 and 22 are each 4 feet from their mean, producing a larger standard deviationHow spread out a set of numbers is around its average.Read the lesson than Line C's.

Solution

Compute each line's variance, since the line with the smallest variance also has the smallest standard deviationHow spread out a set of numbers is around its average.Read the lesson.

Line A is symmetric about 18, so its mean is 18. Its squared deviations from the mean sum to $1(14-18)^2+4(16-18)^2+4(20-18)^2+1(22-18)^2=64$, and there are 10 values, so its variance is $\frac{64}{10}=6.4$.

Line B has mean $\frac{3(14)+3(16)+1(22)}{7}=16$. Its squared deviations sum to $3(14-16)^2+3(16-16)^2+(22-16)^2=48$, and there are 7 values, so its variance is $\frac{48}{7}$.

Line C is symmetric about 18, so its mean is 18. Its squared deviations sum to $2(14-18)^2+(16-18)^2+12(18-18)^2+(20-18)^2+2(22-18)^2=72$, and there are 18 values, so its variance is $\frac{72}{18}=4$.

Line D has mean $\frac{14+22}{2}=18$, and its variance is $\frac{(14-18)^2+(22-18)^2}{2}=16$. Since $4$ is the smallest variance, Line C has the smallest standard deviation.

Report a problem

The values $23, 28, 32, 37, 45$ have a mean of 33. Add one more value, 9.

1. Old total: mean times count.

   $$33\times5=165$$

2. Add the new value.

   $$165+9=174$$

3. Divide by the new count, 6.

   $$174\div6=29$$

Here that's $33+\frac{9-33}{6}=33-4=29$.

1. For the median, the 9 goes first and the old values move up one place.

   $$9,\ 23,\ 28,\ 32,\ 37,\ 45$$

2. Six values, so average the 3rd and 4th.

   $$\frac{28+32}{2}=30$$

The mean fell 4, but the median went only from 32 to 30. The median moves one place at most, however far away the new value is.

To remove a value, take it out of the total and divide by one fewer. The middle places shift back by one.

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The dot plot summarizes weekly new-subscriber counts for a streaming service over 15 weeks. Data set P consists of these counts. Data set Q contains these same counts plus 1,420 new subscribers recorded during a promotional week.

Which of the following correctly compares the means and the medians of data sets P and Q?

1. A

   From P to Q, both the mean and the median increase, but the mean increases more.

2. B

   From P to Q, the mean and the median increase by the same amount.

3. C

   From P to Q, the mean increases, while the median remains unchanged.

4. D

   From P to Q, the mean remains unchanged, while the median increases.

Show the answer

• A

  From P to Q, both the mean and the median increase, but the mean increases more.

  Correct answer

  The mean increases from $460$ to $520$, a gain of $60$. The medianThe middle value once the numbers are put in order.Read the lesson increases from the eighth value, $460$, to the average of the eighth and ninth values, $(460+480)/2=470$, a gain of $10$. Both increase, and the mean increases more.

• B

  From P to Q, the mean and the median increase by the same amount.

  This assumes the high added count increases the medianThe middle value once the numbers are put in order.Read the lesson by the same amount as the mean. The mean increases by $60$, while the change in the middle positions increases the median by only $10$.

• C

  From P to Q, the mean increases, while the median remains unchanged.

  This assumes an added extreme value cannot change the medianThe middle value once the numbers are put in order.Read the lesson. Adding a sixteenth value makes the median the average of the eighth and ninth values, $460$ and $480$, so it rises to $470$.

• D

  From P to Q, the mean remains unchanged, while the median increases.

  This swaps the roles of the two measures, treating the mean as the one an extreme value cannot move. The added week raises the total to $8{,}320$ across $16$ weeks, so the mean rises from $460$ to $8{,}320/16=520$. That is a larger gain than the medianThe middle value once the numbers are put in order.Read the lesson’s rise from $460$ to $470$.

Solution

For data set P, the total is $400+2(420)+4(440)+460+4(480)+2(500)+520=6{,}900$. Its mean is $6{,}900/15=460$. With $15$ values in order, the medianThe middle value once the numbers are put in order.Read the lesson is the eighth value, $460$.

Adding the promotional week gives data set Q a total of $6{,}900+1{,}420=8{,}320$ across $16$ weeks. Its mean is $8{,}320/16=520$. The added value goes at the end of the ordered list, so the eighth and ninth values remain $460$ and $480$. The new median is $(460+480)/2=470$.

The mean increases by $520-460=60$, and the median increases by $470-460=10$. Both increase, but the mean increases more.

Report a problem

Some questions describe a set without listing it. Data set M has 40 whole numbers, each between 300 and 400. Data set N is M plus the number 25. Must the mean be less for N? Must the median?

1. 25 is below every value in M, so N's mean must be less.

   $$25\lt300$$

2. M's median averages its 20th and 21st values.

   $$40 \text{ values}$$

3. N's median is its 21st value. The 25 comes first and pushes M's values up one place, so that's M's 20th.

   $$\frac{41+1}{2}=21$$

If M's 20th and 21st values are equal, the medians are equal. Only the mean must be less.

Adding the same number to every value slides the list along. Add 10 to each value in $2, 5, 9$.

1. Median 5, range 7.

   $$2,\ 5,\ 9$$

2. Median 15, range still 7.

   $$12,\ 15,\ 19$$

Multiplying every value spreads the list out. Multiply each value in $2, 5, 9$ by 3.

1. Median 15, range $27-6=21$.

   $$6,\ 15,\ 27$$

2. The range tripled too.

   $$21=3\times7$$

Some questions add a twist:

• Medians and ranges compared together. Settle the medians first, then the ranges.
• A hidden multiplier. If every value was multiplied by the same number and the mean went from 12 to 48, the number is $48\div12=4$. A range of 9 becomes 36.
• A correction in words. If every value in a table is 4 less than the real amount, add 4 to every value, or add 4 once to the table's mean.
• Two dot plots with the same shape, one farther along the axis, are a shift, so their standard deviations are equal.

Seven values add to 300. With an eighth value, the mean of all eight is 45. What's the eighth?

1. Total for all eight: mean times count.

   $$45\times8=360$$

2. Take away the seven you have.

   $$360-300=60$$

Eleven values add to 560. A 12th value, a whole number strictly between 60 and 75, makes the mean of all 12 a whole number. What's the 12th value?

1. Add 60 and 75 to 560.

   $$620\lt\text{total}\lt635$$

2. The total must divide evenly by 12. 612 is too small and 636 too big.

   $$624=12\times52$$

3. Take away the 11 values you know.

   $$624-560=64$$

Some questions put letters in the list. $a, 14, b, 22, c$ is in order from least to greatest, with median 18, range 30 and mean 20. Find $a$ and $c$.

1. The median is the middle value, and $b$ is in the middle.

   $$b=18$$

2. The range is largest minus smallest.

   $$c-a=30$$

3. The total is mean times count.

   $$a+14+18+22+c=20\times5$$

4. Work out $20\times5=100$, then subtract $14+18+22=54$.

   $$a+c=46$$

5. Add this to $c-a=30$. The $a$ terms cancel.

   $$2c=76$$

6. Divide by 2, then $a=46-38$.

   $$c=38,\ a=8$$

Start with the median, since it hands you a letter for free.

If the letters sit in a frequency table, write the full list with every repeat before you look for the middle place.

Desmos helps with a short printed list. In Bluebook the graph is on top and the lines you type on are below it.

Typing repeats pays for about 20 values or fewer. Skip Desmos for box plots, histograms and sets described in words, and for questions that change every value. The Desmos data and regression lesson goes further.

If two data sets are compared, say in one line what changed.