Averages and Range

A data set is a collection of recorded values. Each value is one observation. If twelve students each report a number of books, there are twelve observations. Two students can give the same answer; both answers belong in the data.

Choose data for a number line

Numerical data are counts or measurements, such as books read or minutes spent reading. Their positions on a number line have meaning. Categorical data are labels, such as favorite sports. A bar graph suits those labels because they have no number-line positions.

Keep every observation

Twelve students report these book counts:

3, 5, 2, 4, 3, 6, 3, 1, 4, 3, 5, 2.

Write the values from least to greatest:

1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6.

The ordered list still has twelve values. The four 3s record four different students. Writing 3 just once would lose three observations. The letter $n$ often stands for the number of observations; here $n=12$.

Build the plot

1. Find the smallest and largest values: 1 and 6 books.
2. Draw a number line covering those values. Space each step equally.
3. Label the line “Books read last month.”
4. Draw one dot for each observation. Stack repeated values above their number.
5. Count all the dots to check the result.

The finished stacks contain:

This table gives the dots to place above each number. On the plot itself, stack them upward. Use equal dot sizes and equal spaces between dots. The total is:

It matches the twelve students. A dot plot, also called a line plot, may use X marks in place of dots. Each mark still records one observation.

Leave room for missing values

Suppose another list contains 4, 5, 9 and 10. A scale with steps of 1 must include 6, 7 and 8, even with no dots there. Jumping straight from 5 to 9 would hide the gap. Equal distances on the number line must show equal changes in value.

A useful final check is to read the data back. One dot above 1 gives one 1; two dots above 2 give two 2s. Reading every stack should recover the complete ordered list.

A dot plot tells you both the recorded values and how often they occur. The number below a stack gives the value. The number of dots gives its frequency, meaning its count.

Read values and counts separately

In the books plot, four dots stand above 3. That means four students each read three books. The height is a count of students; the position is a number of books. To find the number of students in the whole group, count every dot, including dots in shorter stacks.

Words such as “more than” and “at least” change a count:

• “More than 4 books” includes 5 and 6 books: $2+1=3$ students.
• “At least 4 books” includes 4, 5 and 6 books: $2+2+1=5$ students.

Include the boundary value when the wording includes it.

Name what you see

A cluster is a group of values close together. A peak is a tallest stack. A gap is an empty stretch between recorded values. The spread describes how far the values stretch.

For the books plot, most observations lie between 2 and 5 books. The peak is at 3 books. The smallest count is 1 book and the largest is 6. Every whole-number position between them has dots, so there is no gap.

An outlier lies far from the rest. Consider these travel times, in minutes:

6, 8, 8, 10, 10, 10, 12, 12, 14, 15, 40.

Ten times lie between 6 and 15 minutes. The time of 40 minutes stands apart, across a large gap. It is an outlier. Check the record: it could be a typing mistake or a real journey from farther away. Its distance from the others is a reason to investigate, not a reason to erase it.

Describe the whole shape

A symmetric plot has sides that roughly mirror each other. A skewed plot stretches into a longer tail on one side. Name the skew after that tail:

• A tail toward larger values means skewed right.
• A tail toward smaller values means skewed left.

The travel times have a main group near 10 minutes and one far value at 40. That value stretches the whole pattern to the right. The plot is skewed right, even though most dots sit on its left side.

Use numbers and the context in a description. For example: “Most journeys took 6 to 15 minutes. The most common time was 10 minutes. One journey took 40 minutes, far beyond the main group.” This says more than “The dots are mostly together.”

The mode is the value that occurs most often. It can help a store decide which size to stock or show the most popular lunch choice. The mode, median and mean are called averages, or measures of center. They each summarize the data in a different way.

Count the repeats

Shoe sizes sold in one hour were:

6, 9, 7, 5, 7, 8, 6, 7.

An ordered list makes the repeats easy to count:

5, 6, 6, 7, 7, 7, 8, 9.

Size 7 occurs three times. Every other size occurs fewer times, so the mode is size 7. The number 3 is its frequency. When asked for the mode, give the size itself.

You can also tally each size. A tally records one mark per sale, keeping repeated sizes together. The size with the greatest tally is the mode. On a dot plot, read the number under the tallest stack.

Two modes or no mode

In the list 2, 3, 3, 4, 5, 5, 6, both 3 and 5 occur twice. Neither is more frequent than the other, and no value occurs more often. The modes are 3 and 5. A data set with two modes is bimodal. Keep both modes; taking their mean would answer a different question.

In 4, 7, 9, 12, each value occurs once. There is no mode. Write those words. Writing 0 would claim that zero is the most common value, even though zero is absent.

Categories have modes too

A class chooses favorite fruits:

Apple has the largest count, so apple is the mode, or modal category. The answer is the fruit name; 8 is the number of students choosing it. Fruit names have no numerical total or middle position, so the mean and median do not describe these choices.

Common does not always mean central

Consider 1, 1, 8, 9, 10. The mode is 1 because it occurs twice. Yet three of the five values are 8 or more. If you want a typical value near the middle, the mode is a poor guide here. It still correctly answers “What value occurs most often?” The question’s purpose matters when you choose an average.

The median is the middle value of data arranged from least to greatest. Order every observation before looking for the middle. Repeats and zeros each keep their own place.

An odd number of values

The scores 12, 15, 9, 20, 15, 11, 18 become:

9, 11, 12, 15, 15, 18, 20.

There are seven scores. Three come before the fourth score and three after it. The median is 15. The middle of the original, unordered list was 20, which shows why ordering comes first.

One way to locate the middle is to cross off the smallest and largest remaining values together. Continue until one value remains. With seven values, three pairs are crossed off, leaving the fourth value.

An even number of values

Six values have two middle positions. In 3, 5, 8, 10, 12, 14, they are the third and fourth positions. Their values are 8 and 10. Find the number halfway between them:

The median can be absent from the original list. For 4, 6, 7, 9, the middle values are 6 and 7. Their mean is 6.5, so the median can also be a decimal.

Positions are not values

For an ordered list with $n$ observations, the middle position is:

For seven observations this gives position 4. Read the value in that position to get the median. For twenty observations it gives 10.5: halfway between positions 10 and 11. Find those two values and take their mean. The position number 10.5 is not itself the median.

Repeated values still occupy separate positions. In 0, 0, 2, 3, 7, the median is 2. Removing one zero would change the list and its middle.

Dots and decimals

Dots on a number line are already arranged by value. Count individual dots from the left, including each dot in a stack. In the twelve-student books plot, positions 6 and 7 both fall above 3. The median is 3 books.

For running times of 12.4, 11.8, 13.1 and 12.0 seconds, first order them:

11.8, 12.0, 12.4, 13.1.

The two middle times give:

The median is 12.2 seconds. Its unit stays the same as the unit of each recorded time.

The range tells how far a set of numbers stretches. It is a measure of spread. Find the largest observed value and subtract the smallest:

Find the ends of the data

Daily high temperatures were 71, 66, 80, 75 and 68 °F. The lowest was 66 °F and the highest was 80 °F:

The range is 14 °F. Saying “66 to 80 °F” gives the endpoints. The range is the single number measuring their difference.

The first and last entries of an unordered list may be anywhere in the data. Here they are 71 and 68. Subtracting them would miss both extremes. Ordering the list helps you find the true smallest and largest values.

A range cannot be negative: the largest value is at least as large as the smallest. If every result is 7 points, the range is zero. All the results coincide.

Read observed values on a dot plot

Use the leftmost and rightmost dots, not the printed ends of the scale. A number line may run from 0 to 10 while the dots occupy 1 through 6. Its range is:

The empty positions beyond the dots add no observations. Gaps inside the data remain part of the distance between the extremes.

Explain a small or large range

One player’s scores lie between 12 and 18 points. Another player’s lie between 6 and 23 points. Their ranges are 6 and 17 points. The first player’s scores are closer together, so they are more consistent over those games.

A large range describes a large spread. It does not mean that the scores themselves are high. Two sets can have similar averages while one has much more spread.

Watch for an outlier

The travel times 6, 8, 8, 10, 10, 10, 12, 12, 14, 15 and 40 minutes have a range of:

Without the time of 40 minutes, the range would be:

One far value changes the range greatly because the rule uses only the two extremes. Moving a value inside the existing endpoints leaves the range unchanged.

The mean is the total of all values divided by their count. In everyday speech, “the average” usually means the mean.

Share without changing the total

Four friends have 3, 5, 6 and 10 stickers. Their total is:

If they pool the stickers and share them equally, each receives:

Their mean is 6 stickers. Only one friend started with six, but the mean describes what each would have after equal sharing.

You can picture this by leveling cube towers. Towers of 2, 7 and 3 cubes contain twelve cubes altogether. Move two cubes from the tallest tower to the first tower. Move one more to the third tower. Each tower now has four cubes. Nothing was added or lost; the mean is four cubes.

Find the balance point

The mean also balances distances on a number line. For the sticker counts, compare each value with 6:

The distances below total 4, matching the distance above. The mean is a balance point. Values farther away have a greater pull, just as a load farther from a seesaw’s center has more pull.

Check what the mean can be

The mean lies between the smallest and largest values, including the endpoints. A mean of 12 for 3, 5, 6 and 10 stickers would fail that check. Equal sharing cannot give everyone more than the largest starting amount.

The mean need not occur in the data. Families with 1, 2, 3 and 4 children have ten children altogether. The mean is 2.5 children per family. That describes the group’s total shared across four families. It does not describe part of an actual child.

Zero observations count too. Three students score 0, 4 and 5 goals. The total is nine goals, shared across three students, so the mean is three goals. Dividing by two would leave out the student with no goals.

Three averages can give different answers

For 2, 3, 3, 4, 8, the total is 20 and the mean is 4. The median is 3 and the mode is also 3. The mean shares the total; the median locates the ordered middle; the mode finds the most common value. You can get different answers while calculating all three correctly.

To calculate a mean, add every value, count the observations, and divide. For quiz scores of 7, 9, 6, 10 and 8, the total is 40 points. Five scores give a mean of 8 points. Check that the result lies between 6 and 10.

Decimals and calculator entry

Five daily rainfall measurements are 1.2, 0.8, 0, 2.5 and 1.5 inches. Their total is 6 inches, so the mean is 1.2 inches per day. The dry day contributes zero to the total and one to the count.

For 5, 8, 6, 9, 7, 4, 8, the total is 47. The mean is:

Round only the final result. To one decimal place, the mean is 6.7. Follow a question’s requested accuracy; otherwise use one decimal place when rounding is needed.

A calculator must divide the whole sum. Find the sum and press equals before dividing, or enter:

Without parentheses, an entry such as $5+8+6+9+7+4+8÷7$ divides only the last 8. Division happens before addition, giving about 40.1. That is above every original value, so it cannot be their mean.

In a spreadsheet, put one observation in each cell. For cells A1 through A7, use =AVERAGE(A1:A7) for the mean, =MEDIAN(A1:A7) for the median, and =MAX(A1:A7)-MIN(A1:A7) for the range.

Work backward to the total

Reversing the division gives:

Five test scores with mean 82 points must total:

Use the count that belongs to the mean.

Find a missing value

Five numbers have mean 8. Four are 6, 9, 10 and 11.

1. The required total is $8\times 5=40$.
2. The known total is $6+9+10+11=36$.
3. The missing value is $40-36=4$.

Check by adding all five numbers and dividing by five.

Reach a target mean

Maya’s first four test scores have mean 80. She wants mean 82 after five tests. The old and target means belong to different counts:

• First four tests: $80\times 4=320$ points.
• All five tests: $82\times 5=410$ points.
• Needed fifth score: $410-320=90$ points.

A score of 82 on the last test would not undo the lower earlier mean.

Combine groups of different sizes

One class has 20 students with mean score 15. Another has 10 students with mean score 18. Recover each total:

Together they have 480 points across 30 students:

The combined mean is 16 points. Averaging 15 and 18 gives 16.5, but that treats the two classes as equally large. The larger class contains more scores, so it contributes more to the total.

A frequency table pairs each value with its count. The value tells what happened in one observation. The frequency tells how many observations had that result.

Read each row as a sentence

A soccer team records its goals in twenty games:

Read the first row as “Four games had zero goals.” Read the row for 2 as “Five games had two goals each.” The frequency of 5 counts games. It is not a score of five goals.

The table could be expanded into a list: four 0s, seven 1s, five 2s, three 3s and one 4. A dot plot would show stacks of those same heights. Each display holds the same observations.

Count the observations

Add the frequencies to find the number of games:

The total frequency is often called $n$. Tally marks can help collect these frequencies: make a mark for each game in its matching goals row, then count the marks.

Adding the goal labels gives $0+1+2+3+4=10$. That sum counts neither the games nor the team’s total goals. A row label must be repeated as many times as its frequency says.

Find the mode

The highest frequency is 7. It belongs to 1 goal, so the mode is 1 goal. The table says a one-goal game occurred more often than any other result. Reporting 7 would give the number of those games instead of their score.

The same reading works for categories. In a favorite-fruit table, find the largest frequency and report its fruit name as the mode. Categories such as apple and banana have no numerical range.

Find the range

The smallest observed goal count is 0. The largest is 4:

The range is 4 goals. Subtract the observed goal counts, not the frequencies of their rows.

Suppose a further row lists 5 goals with frequency 0. It records no five-goal games. The largest observed score remains 4, so the range stays 4 goals. A printed label only becomes part of the data when its frequency is greater than zero.

A table’s median is the middle of all its observations. Rows can hold different numbers of observations. To locate the middle, put the values in order and track the positions filled by each row.

Add frequencies as you go

A running total, also called cumulative frequency, adds each frequency to those above it. For the soccer table:

The first four entries of the full list are 0. Adding the seven one-goal games takes the count to 11. Those 1s fill positions 5 through 11. Each running total gives the last position filled by that row.

Locate both middle positions

Twenty games give an even count. The middle lies between positions 10 and 11. Both are in the one-goal row, so both middle values are 1. The median is 1 goal.

The formula $\frac{n+1}{2}$ gives position 10.5 here. It points halfway between the two positions. It does not say that the team’s median score is 10.5 goals.

An odd total

Another table records people per car:

There are 25 cars, so the middle is position 13. The first row ends at position 9. The next row fills positions 10 to 17, including 13. The median is 2 people per car.

A middle pair can cross rows

Sixteen students report books read: 1 book for 3 students, 2 for 5 students, 3 for 4 students, and 4 for 4 students. The running totals are 3, 8, 12 and 16.

The middle positions are 8 and 9. Position 8 is the last 2; position 9 is the first 3. Their mean gives:

There are four rows, but the median is not found by taking the middle of those row labels alone. Each label represents several students. For a small table, expand it into a full ordered list to check that you reach the same middle values.

The mean always shares the total across the observations. In a frequency table, multiplication rebuilds the total that the shorter display hides.

Multiply within each row

Five games with two goals each contribute ten goals. The row calculation is:

Multiply each value by its frequency. Add a product column to the soccer table:

The final column counts goals. The frequency column counts games. The team scored thirty goals in twenty games, giving:

The mean is 1.5 goals per game. No single game has half a goal. This is the team’s total shared equally across all its games.

Use the right denominator

The rule is:

Some books write $\frac{\mathrm{\Sigma }fx}{\mathrm{\Sigma }f}$. Here $x$ is a value, $f$ is its frequency, and $\mathrm{\Sigma }$ means “add them all.” The top adds the products; the bottom adds the frequencies.

Dividing thirty by five rows gives six. That cannot be the mean because the scores only go from zero to four. Averaging the values column alone also fails: it treats every score as equally frequent, although one-goal games occurred seven times and four-goal games only once.

A zero value still has a frequency. The four scoreless games add zero goals but remain four of the twenty games.

Try a different unit

For cars with 1 person in 9 cars, 2 people in 8 cars, 3 people in 5 cars and 4 people in 3 cars, the products count people:

There are 25 cars. The mean is:

That is 2.08 people per car, between the smallest and largest counts.

The soccer table now gives all four summaries: mode 1 goal, median 1 goal, mean 1.5 goals per game and range 4 goals. The few larger scores raise the mean above the median. The method changes with the measure, even though the data stay the same.

The mean uses every value’s size. The median uses the middle positions. The mode uses counts, and the range uses the two extremes. These different rules explain why changing the data affects each measure differently.

Follow one outlier

Travel times of 6, 8, 8, 10, 10, 10, 12, 12, 14, 15 and 40 minutes include one unusually long journey. Compare the summaries with and without that 40-minute time:

The full total is 145 minutes. Without 40 it is 105. Both the total and the count change when a value is removed:

The large time pulls the mean upward. The middle stays at 10, and 10 remains most frequent. The range changes greatly because its largest value changes. Removing an observation can move the median in other lists; check the new middle positions.

Check unusual records

An outlier can be real. A student may live much farther from school. Keep a genuine observation and explain its effect. If a record says 400 when the actual time was 40, correct the error. If the true value cannot be recovered, remove the faulty record and report why. Deleting a real result merely because it changes the average gives a misleading picture.

Add or remove one value

Compare a new value with the current mean:

• Above the mean: adding it raises the mean.
• Below the mean: adding it lowers the mean.
• Equal to the mean: adding it leaves the mean unchanged.

The quiz scores 7, 9, 6, 10, 8 have mean 8. Adding another 8 gives 48 points across six scores, still mean 8. Adding 14 instead gives 54 points across six scores, mean 9.

Removal reverses the direction. Removing an above-mean value lowers the mean of the remaining data. Removing a below-mean value raises it.

Change every value together

If every quiz score gains two points, every average gains two points. The smallest and largest scores also both rise by two, so their difference stays fixed:

The range remains 4 points.

Link averages to shape

A long right tail usually pulls the mean above the median. A long left tail usually pulls it below the median. In roughly symmetric data, the two are close. A large difference is a reason to inspect the plot for a tail or an outlier.

An average should help answer a question about the data. Start by asking what you need: the most common result, a typical middle value, or an equal share of the total.

Choose by purpose

For favorite lunch choices, the mode identifies the most popular category. Adding sandwich names or finding their numerical middle has no meaning.

A store choosing which shoe size to order also needs the most common size. A mean size such as 6.875 does not identify the size sold most often. The numerical data still call for a mode because the purpose is stocking a popular size.

Use the median for an uneven group

Seven children receive these weekly allowances:

$5, $5, $6, $8, $10, $10, $45.

The total is 89 dollars. The mean is about $12.71, more than six of the seven children receive. The large allowance pulls the mean upward. The median is $8, a more useful description of a typical child’s allowance here.

The modes are $5 and $10. Both are common, but neither gives the ordered middle. A clear reason for choosing the median is: “The $45 allowance is an outlier, so the median of $8 better describes the typical allowance.”

The median is also useful for typical house prices or incomes when a few very large amounts stretch the data upward. When reading “average,” check which average the writer means.

Use the mean when the total matters

For the fairly balanced quiz scores 7, 9, 6, 10, 8, the mean of 8 gives a useful summary using all five scores. The median works too. If the task is to share a cost equally, the mean directly gives each person’s share, because everyone’s share must add back to the total.

Use this order to make a choice:

1. For unordered categories, use the mode.
2. For a typical numerical value, inspect the shape and outliers. An outlier or long tail favors the median.
3. For fairly symmetric numerical data without outliers, the mean is usually useful.
4. Check the purpose. A question about equal sharing still needs the mean; a question about the most common size needs the mode.

A reason should name a feature of these data or this purpose. “The median is best” leaves that reasoning unexplained.

An average describes a group’s center. The range describes its spread. Comparing both helps you see how high or low results are and how closely they agree.

Calculate the same measures

Ana’s six game scores are 12, 14, 15, 15, 16 and 18 points. Ben’s are 6, 10, 15, 20, 22 and 23 points.

For Ana, divide 90 by six games to get 15. Subtract 12 from 18 to get 6. Ben’s corresponding calculations give 16 and 17.

Compare mean with mean, or median with median. Mixing Ana’s mean with Ben’s median would compare two different kinds of summary.

Say what the numbers mean

“Ben scored more points per game on average: his mean was 16 points, compared with Ana’s 15. Ana’s scores were more consistent: her range was 6 points, compared with Ben’s 17.”

Each sentence includes both results and their meaning. A table of numbers alone leaves the reader to make the comparison.

Ben’s larger range means a wider spread. It does not mean he scored more points; his mean supplies that separate information. A broad claim such as “Ben is better” needs a purpose. Ben has the higher scoring average in these games. Ana is steadier by range. Six games also give limited evidence about later performance.

Decide what higher means

Higher is usually better for points scored. Lower is better for race times or numbers of errors. A swimmer with mean time 31.2 seconds is faster on average than one with mean time 32.5 seconds. Keep the unit and the meaning together before judging a result.

Compare groups of different sizes

A team scores 60 points in five games. Another scores 66 points in six games. Their means are:

The first team scores more per game on average, despite its smaller total. Totals alone mix performance with the number of games played.

Compare dot plots fairly

Draw both plots on the same scale. A group farther to the right has larger values; a wider group has more spread. Match that picture to the calculated average and range. If the scales differ, compare the labeled values carefully: the width on the page alone can be misleading.