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Mean, Median, Mode, Range, and Weighted Averages

Find and choose among the mean, median, mode, and range; find a missing data value; and compute weighted averages.

Practice this skill 28 min Mean, median, mode, range, weighted average Leer en español
Key idea

What you'll learn

When you have a list of numbers — paychecks, test scores, gas prices — you often want one number that describes the whole list. In this lesson you will learn the four measures the GED uses:

  • Mean (the average),
  • Median (the middle),
  • Mode (the most common), and
  • Range (the spread).

You will also learn to:

  • choose the best measure of center when data have an outlier,
  • work backwards to find a missing value when you know the mean ("What do I need on the next test?"), and
  • compute a weighted average, where some values count more than others (course grades, average prices).

Why it matters on the GED: Measures of center appear on almost every GED Math test, often several times. Questions range from a plain "find the mean" to two- and three-step problems like finding a missing score or a weighted grade. Outside the test, you use these ideas to compare prices, understand your grade in a class, judge a typical salary in a job ad, or track your monthly bills.

Words to know

  • Data set — a list of numbers, such as 12, 15, 9, 18, 21.
  • Measure of center — one number that describes a typical value (mean, median, or mode).
  • Mean (average) — the sum of the values divided by how many values there are. (Mean of 2, 4, 9: \tfrac{15}{3} = 5.)
  • Median — the middle value when the data are in order. (Median of 9, 2, 4 → order 2, 4, 9 → 4.)
  • Mode — the value that appears most often. (Mode of 3, 5, 5, 8: 5.) A set can have no mode or more than one.
  • Range — the greatest value minus the least value. (Range of 2, 4, 9: 9 - 2 = 7.)
  • Measure of spread — a number that tells how spread out the data are (range).
  • Outlier — a value far from the others (150 in 32, 35, 35, 38, 40, 150).
  • Weighted average — an average where some values count more than others (tests 50%, homework 30%).
  • Weight — how much a value counts, as a percent, decimal, or count.
  • Sum (\sum) — the result of adding values.

All Math words in the glossary

Mean, median, mode, and range

Mean. Add all the values, then divide by the number of values.

\text{mean} = \frac{\text{sum of values}}{\text{number of values}}

Example: 12, 15, 9, 18, 21 → sum 75, count 5, mean = 75 \div 5 = 15.

Think of the mean as a fair share: if the total were split evenly, each value would be the mean. The mean can be a decimal even when the data are whole numbers: the mean of 3, 4, 4 is \tfrac{11}{3} \approx 3.67.

Mean as a fair share. Bars for 12, 15, 9, 18 and 21 with a red line at the mean, 15. The parts of the bars above the line (3 from 18 and 6 from 21, total 9) exactly fill the gaps below the line (3 under 12 and 6 under 9, total 9), so leveling all bars gives 15 each. Sum 75 ÷ 5 = 15.03691215182124−31215−69+318+621mean 15extra above (3 + 6) fills gaps below (3 + 6)sum 75 ÷ 5 values = 15 each

The mean is also the balance point: the distances below it and above it cancel out.

Mean as a balance point. The values 9, 12, 15, 18 and 21 are dots on a number line resting on a triangle at the mean, 15. Distances from the mean are −6, −3, 0, +3 and +6; the negatives and positives cancel (−9 + 9 = 0), so the line balances at 15.8910111213141516171819202122−6−30+3+6balance point = mean = 15distances: (−6) + (−3) + 0 + 3 + 6 = 0

Median. Put the data in order, then find the middle.

  • Odd number of values: the median is the single middle value. Example: 4, 7, 9, 12, 20 → median 9.
  • Even number of values: average the two middle values.
Eight values in order from smallest to largest: 25, 28, 30, 32, 36, 38, 40, 40. The two middle values, 32 and 36, are highlighted; the median is their average, 34.2528303236384040middlesmallest → largest

Example: 32, 40, 28, 36, 40, 25, 38, 30 → sorted 25, 28, 30, 32, 36, 38, 40, 40 → median = \tfrac{32 + 36}{2} = 34.

For n values, the median is in position \tfrac{n + 1}{2}. For n = 8, that's 4.5, meaning halfway between the 4th and 5th values.

Mode. The value that appears most often. Example: 3.29, 3.45, 3.19, 3.45, 3.39, 3.29, 3.45 → $3.45 appears 3 times, so the mode is $3.45. If no value repeats, there is no mode. If two values tie for most, both are modes.

Range. Greatest minus least. Example: gas prices: 3.45 - 3.19 = \$0.26. Range is always zero or positive.

Negatives and decimals work the same way. Example: temperatures -4, 2, -1, 5, 3 (°F). Sum = 5, mean = 5 \div 5 = 1°F. Sorted: -4, -1, 2, 3, 5, median = 2. Range = 5 - (-4) = 9°F.

Temperatures −4, −1, 2, 3 and 5 degrees F on a number line. The median is the middle value, 2 (red). The mean is 1 (balance triangle): sum 5 ÷ 5 = 1. The range bracket from −4 to 5 is 5 − (−4) = 9 degrees.-5-4-3-2-10123456median 2mean 1range = 5 − (−4) = 9°Fsorted: −4, −1, 2, 3, 5; sum = 5

Choosing the best measure; outliers

Each measure answers a different question:

MeasureBest when…Affected by outliers?
Meandata are balanced, no extreme valuesYes, strongly
Mediandata have outliers or are skewed (salaries, home prices)Hardly at all
Modeyou want the most common item (most popular shoe size)No
Rangeyou want the spreadYes — it uses the extremes

Example: Salaries (thousands): 32, 35, 35, 38, 40, 150.

  • Mean = 330 \div 6 = 55 → $55,000.
  • Median = \tfrac{35 + 38}{2} = 36.5 → $36,500.

Five of six people earn less than the mean, so the mean doesn't describe a typical worker. The median does. A high outlier pulls the mean above the median; a low outlier pulls it below.

Salaries in thousands of dollars on a number line from 0 to 160: 32, 35, 35, 38, 40 and an outlier at 150. The median, 36.5, sits among the five typical salaries. The mean, 55, is pulled far to the right by the outlier, above five of the six salaries.020406080100120140160outlier 150median 36.5mean 55 (pulled toward the outlier)Salary (thousands of $)

How changes affect the measures:

  • Add the same amount to every value (everyone gets a $2,000 raise): the mean, median, and mode all go up by that amount; the range stays the same.
  • Remove an outlier: the mean changes a lot; the median changes a little. Removing $150,000 drops the mean from $55,000 to \tfrac{180}{5} = \$36{,}000, but the median only moves from $36,500 to $35,000.
  • Remember: adding or removing a value changes the count, so always recompute.

Missing values and weighted averages

Finding a missing value (work backwards). The key fact:

\text{sum} = \text{mean} \times \text{number of values}

Example: Scores 82, 75, 90. What 4th score gives a mean of 84?

  1. Needed total: 4 \times 84 = 336.
  2. Current total: 82 + 75 + 90 = 247.
  3. Missing score: 336 - 247 = 89.
Tape model for the missing score. The needed total is 4 × 84 = 336. The tape of 336 is split into the known scores 82, 75 and 90 (sum 247) and an unknown piece, 336 − 247 = 89.needed total = 4 × 84 = 336827590? = 89current total 247336 − 247

Example (means only): The mean of 6 scores is 85; after a 7th test, the mean is 86. The totals are 6 \times 85 = 510 and 7 \times 86 = 602, so the 7th score is 602 - 510 = 92.

Weighted average. Use this when values don't count equally.

Type 1 — weights are percents (adding to 100%): multiply each value by its weight as a decimal and add. Tests 50% (82), homework 30% (95), final 20% (74):

0.5(82) + 0.3(95) + 0.2(74) = 41 + 28.5 + 14.8 = 84.3
Weighted grade as a tape of 100% split into tests 50%, homework 30% and final 20%. Each piece earns its score times its weight: 0.5 × 82 = 41, 0.3 × 95 = 28.5, 0.2 × 74 = 14.8. Added together, 41 + 28.5 + 14.8 = 84.3.Tests 50%score 820.5 × 82= 41Homework 30%score 950.3 × 95= 28.5Final 20%score 740.2 × 74= 14.8grade = 41 + 28.5 + 14.8 = 84.3

Type 2 — weights are counts: multiply each value by its count, add, then divide by the total count. 40 shirts at $12, 25 at $18, 15 at $25:

\frac{12(40) + 18(25) + 25(15)}{40 + 25 + 15} = \frac{1{,}305}{80} \approx \$16.31

The simple mean of the three prices, \tfrac{12 + 18 + 25}{3} \approx \$18.33, is wrong here because more shirts sold at the low price.

Weighted average of shirt prices. Stacks on a number line show 40 shirts at 12 dollars (tallest), 25 at 18 dollars and 15 at 25 dollars, with heights proportional to the counts. The weighted mean, about 16.31 dollars, is the balance point (blue triangle), pulled toward 12 dollars. The simple mean of the three prices, 18.33 dollars, is marked with a red dashed line and is too high.$10$12$14$16$18$20$22$24$2640 shirts25 shirts15 shirtsweighted mean ≈ $16.31simple mean $18.33 (wrong)= 1,305 ÷ 80

A weighted average always lies between the smallest and largest values, and it is pulled toward the values with the bigger weights.

Formula

Formulas (what is and isn't on the sheet)

\text{Mean} = \frac{x_1 + x_2 + \cdots + x_n}{n} \text{Range} = \text{greatest value} - \text{least value} \text{Median position} = \frac{n + 1}{2} \quad \text{(after sorting)} \text{Missing value} = (\text{target mean} \times n) - (\text{sum of the known values}) \text{Weighted average} = \frac{w_1x_1 + w_2x_2 + \cdots + w_kx_k}{w_1 + w_2 + \cdots + w_k}
  • x_1, x_2, \ldots — the data values; n — how many values there are
  • w_1, w_2, \ldots — the weights (percents as decimals, or counts). When percent weights add to 1, the denominator is 1 and you just add the products.

On the GED formula sheet: the definitions of mean (total of the values ÷ number of values) and median (middle value of an ordered set; mean of the two middle values if the count is even).

Not on the sheet — memorize: mode, range, the missing-value method and the weighted-average formula. They come up on nearly every test.

Worked example

Example 1 (easy): All four measures

Find the mean, median, mode, and range of: 7, 3, 9, 3, 8.

  1. Sort: 3, 3, 7, 8, 9.
  2. Mean: \dfrac{3 + 3 + 7 + 8 + 9}{5} = \dfrac{30}{5} = 6.
  3. Median: 5 values, middle (3rd) value = 7.
  4. Mode: 3 (appears twice).
  5. Range: 9 - 3 = 6.
Example 1. The data 3, 3, 7, 8, 9 as dots on a number line from 2 to 10. Mode 3 is the stacked pair of dots. Median 7 is the middle dot (red). Mean 6 is the balance point (green triangle). The range bracket from 3 to 9 is 6.2345678910mode 3median 7mean 6 = 30 ÷ 5range = 9 − 3 = 6

Answer: mean 6, median 7, mode 3, range 6.

Check: Mean \times count = 6 \times 5 = 30 = the sum ✓

Worked example

Example 2 (medium): Monthly bills — real world

Maya's electric bills for six months were $84, $92, $78, $110, $96, and $80. What are the mean and median bill?

  1. Sum: 84 + 92 + 78 + 110 + 96 + 80 = 540.
  2. Mean: 540 \div 6 = \$90.
  3. Sort: 78, 80, 84, 92, 96, 110. Middle two: 84 and 92.
  4. Median: \dfrac{84 + 92}{2} = \$88.
Example 2. Bar graph of Maya's six electric bills: 84, 92, 78, 110, 96 and 80 dollars. A red line marks the mean, 90 dollars (540 ÷ 6), and a green dashed line marks the median, 88 dollars. The 110 dollar bill pulls the mean above the median.020406080100120Bill ($)$84Mo 1$92Mo 2$78Mo 3$110Mo 4$96Mo 5$80Mo 6mean $90median $88Month

Answer: mean $90, median $88.

Check: The $110 bill is the highest and pulls the mean a little above the median ✓

Worked example

Example 3 (medium): Find the missing value

A basketball player scored 18, 24, 15, and 21 points in four games. How many points must she score in the fifth game to average 20 points per game?

  1. Needed total: 5 \times 20 = 100.
  2. Current total: 18 + 24 + 15 + 21 = 78.
  3. Missing value: 100 - 78 = 22.
Example 3. Bars for points in five games: 18, 24, 15, 21, and the unknown fifth game, 22 (dashed red). A red line marks the target mean, 20. Needed total 5 × 20 = 100; current total 78; missing value 100 − 78 = 22.0481216202428Points18G124G215G321G4?G5mean 20need: 5 × 20 = 100have: 78? = 100 − 78= 22

Answer: 22 points.

Check: \dfrac{78 + 22}{5} = \dfrac{100}{5} = 20 ✓

Worked example

Example 4 (GED-hard): Weighted average with counts

A survey asked 50 families how many cars they own: 8 own 0 cars, 20 own 1, 15 own 2, and 7 own 3. What is the mean number of cars per family?

  1. Multiply each value by its count: 0(8) = 0, 1(20) = 20, 2(15) = 30, 3(7) = 21.
  2. Total cars: 0 + 20 + 30 + 21 = 71.
  3. Total families: 8 + 20 + 15 + 7 = 50.
  4. Mean: 71 \div 50 = 1.42.
Example 4. Bar graph of how many cars 50 families own: 0 cars 8 families, 1 car 20, 2 cars 15, 3 cars 7. The mean, 71 ÷ 50 = 1.42 cars, is the balance point marked by a green triangle under the axis, pulled toward 1 where most families are. The unweighted 1.5 is marked with a red dashed line for comparison.048121620Number of families80 cars201 car152 cars73 carsmean 1.42 = 71 ÷ 501.5 ignores counts (wrong)

Answer: 1.42 cars per family.

Check: The answer is between 0 and 3, and close to 1 because more families own 1 car than any other number ✓ (The wrong method, \tfrac{0 + 1 + 2 + 3}{4} = 1.5, ignores the counts.)

Worked example

Example 5 (GED-hard): Work backwards with a weighted grade

A class grade is 70% tests and 30% project. Jordan's project score is 95. What test average does he need for a final grade of exactly 88?

  1. Let T be the test average: 0.7T + 0.3(95) = 88.
  2. Simplify: 0.7T + 28.5 = 88.
  3. Subtract: 0.7T = 59.5.
  4. Divide: T = 59.5 \div 0.7 = 85.
Example 5. A tape for the final grade of 88 split into the tests part, 70% (0.7T) and the project part, 30% (0.3 × 95 = 28.5). The tests part must be 88 − 28.5 = 59.5, so T = 59.5 ÷ 0.7 = 85.final grade = 88Tests 70%: 0.7T = ?Project 30%0.3 × 95 = 28.50.7T = 88 − 28.5 = 59.5T = 59.5 ÷ 0.7 = 85

Answer: a test average of 85.

Check: 0.7(85) + 0.3(95) = 59.5 + 28.5 = 88 ✓

Calculator tip

Using the TI-30XS MultiView

The calculator is very helpful for means and weighted averages:

  • Mean: put the sum in parentheses, then divide: ( 12 + 15 + 9 + 18 + 21 ) ÷ 5 enter → 15. Without the parentheses, the calculator divides only the last number by 5 (order of operations) and gives a wrong answer.
  • Weighted average with percents: 0.5 × 82 + 0.3 × 95 + 0.2 × 74 enter → 84.3. Multiplication happens before addition, so no parentheses are needed.
  • Weighted average with counts: ( 12 × 40 + 18 × 25 + 25 × 15 ) ÷ ( 40 + 25 + 15 ) enter → 16.3125.
  • Negative values: type them with the (−) key, e.g. (−) 4.
  • If a result shows as a fraction, press toggle to see the decimal.

When not to use it: finding the median and mode is about sorting and counting — do it on your erasable note board. For small whole-number lists (like 3, 3, 7, 8, 9), mental math is often faster.

Common mistake

Common mistakes

  • Not sorting before finding the median. Wrong: middle of 32, 40, 28, 36, 40, 25, 38, 30 → 38. Right: sort first → 25, 28, 30, 32, 36, 38, 40, 40 → median 34.
  • Forgetting to divide (or dividing by the wrong count). Wrong: mean = 75 (just the sum) or 75 \div 4. Right: divide the sum by the number of values, 75 \div 5 = 15.
  • Range = largest value. Wrong: range of 6, 9, 9, 10, 11, 15 is 15. Right: 15 - 6 = 9.
  • Ignoring weights. Wrong: grade = \tfrac{82 + 95 + 74}{3} \approx 83.7. Right: 0.5(82) + 0.3(95) + 0.2(74) = 84.3.
  • Averaging averages. Wrong: mean after the 7th test is between 85 and 86, so the score is 85.5. Right: use totals: 7 \times 86 - 6 \times 85 = 92.
  • Using the mean when there is an outlier. Wrong: "typical salary is $55,000." Right: the median, $36,500, is more typical.
Test strategy

GED test strategy

  • Read the question twice and underline which measure it asks for: mean, median, mode, or range. The wrong choices are usually the other three measures, already worked out for you.
  • Sort the data on your erasable note board before any median, mode, or range question.
  • Estimate first. The mean and median must be between the smallest and largest values. Cross out any choice outside that range.
  • Plug answers back in for missing-value questions: add the choice to the list and check the mean.
  • Use sum = mean × count whenever a mean is given and something is missing — it's the fastest route.
  • For weighted averages, check that the weights add to 100% (or find the total count) before you start.
  • Time: a basic mean or median should take under a minute. Save extra time for multi-step weighted-average questions.

Check yourself

Mean, median, mode, range, weighted average

Over five days, a food truck sold 14, 19, 11, 23, and 18 burritos. What is the mean number of burritos sold per day?

Answer choices
Hint

The mean is the "fair share": add all the values, then divide by how many values there are.

Check yourself

Mean, median, mode, range, weighted averageFill in

Luis worked these numbers of hours over 8 weeks:

35, 42, 28, 45, 40, 33, 38, 30

What is the median number of hours?

hours

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Put the numbers in order first. With an even number of values, there are two middle numbers.

Summary

  • Mean = sum ÷ count; median = middle of the sorted data (average the two middle values if the count is even).
  • Mode = most frequent value; range = greatest − least.
  • Outliers pull the mean strongly; the median is better for skewed data such as salaries.
  • Missing value = target mean × count − known sum.
  • Weighted average = \dfrac{\sum (\text{weight} \times \text{value})}{\sum \text{weights}}; with percent weights, just add the products.
  • Mean and median are defined on the GED formula sheet; mode, range, missing value and weighted average are not — memorize those.
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