ANTMACK215 × WEALTHWISEGUYS
WealthWiseGuys Practical Mathematics

Chapter 22

Business Statistics, Probability, and Data Interpretation

Averages, chance, risk, and how charts can mislead

Box scores, odds tickers, creator dashboards, and stock apps all throw numbers at you. This chapter teaches you to summarize data, measure chance, see why the house wins, and catch a chart that is trying to fool you. All teams, players, prices, and odds are hypothetical.

22.1

Mean, Median, Mode, and Range

Learning objectives

After this section you will be able to:

  1. Find the mean, median, mode, and range of a data set.
  2. Find the median when there is an even number of values.
  3. Decide which average best describes data with an outlier.

Real-life situation

A guard's last seven games

A high-school guard wants to tell a college recruiter what she "usually" scores. Her last seven games are listed in the stat sheet. Different averages give different answers, and she needs to know which one is fair.

Game1234567
Points18241230241531
Figure 22.1 Points scored, last seven games (hypothetical player)

The concept

A statistic is a single number that summarizes a set of data. The three most common "averages" are called measures of center.

The mean is what most people call the average: add every value, then divide by how many values there are. The median is the middle value after you put the data in order. If there is an even number of values, the median is the mean of the two middle values. The mode is the value that appears most often; a data set can have one mode, several, or none.

The range measures spread, not center: it is the largest value minus the smallest. A large range means the results vary a lot from game to game.

An outlier is a value far away from the rest. Outliers pull the mean toward them but barely move the median, so the median is often the fairer "typical" value for incomes, prices, and earnings.

Distribution of seven game scores

101520253035
  1. Read the dot plot: repeated scores stack vertically.
  2. Sort: 12, 15, 18, 24, 24, 30, 31. Median = 24; mode = 24.
  3. Mean = 22 points; range = 19. Different summaries describe different features.

Original WealthWiseGuys diagram · Hypothetical teaching quantities.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • mean vs. median

    The mean is the arithmetic average; the median is the middle value. Outliers pull the mean more than the median.

  • outlier

    A value far away from the rest of the data, which can make an average misleading.

  • risk

    The chance that an investment or decision will turn out worse than expected.

Key terms

Data set
A collection of numbers about the same thing, such as points per game.
Mean
The sum of the values divided by the number of values.
Median
The middle value of the ordered data.
Mode
The value that occurs most often.
Range
Largest value minus smallest value.
Outlier
A value much larger or smaller than the others.

Formulas & rules

  • Mean

    Mean = (sum of values) ÷ (number of values)

  • Range

    Range = maximum − minimum

  • Median (even count)

    Median = (middle value 1 + middle value 2) ÷ 2

    Always sort the data first.

Worked examples

Example 1The guard's stat sheet

Find the mean, median, mode, and range of 18, 24, 12, 30, 24, 15, 31.

Solution

  1. 1Sum: 18 + 24 + 12 + 30 + 24 + 15 + 31 = 154.
  2. 2Mean: 154 ÷ 7 = 22 points.
  3. 3Order the data: 12, 15, 18, 24, 24, 30, 31. The 4th of 7 values is the middle, so the median is 24.
  4. 4Mode: 24 appears twice; every other value once. Mode = 24.
  5. 5Range: 31 − 12 = 19 points.

Answer: Mean 22, median 24, mode 24, range 19

Example 2A creator's payouts with an outlier

A video creator's monthly payouts were $40, $55, $35, $60, and $910 (one video went viral). Which average better describes a typical month?

Solution

  1. 1Mean: (40 + 55 + 35 + 60 + 910) ÷ 5 = 1,100 ÷ 5 = $220.
  2. 2Order: 35, 40, 55, 60, 910. Median = $55.
  3. 3Four of five months were $60 or less, so $220 overstates a normal month. The $910 outlier pulled the mean up.

Answer: The median, $55, is the better description of a typical month.

Example 3Median of an even-sized set

Find the median of 3, 8, 10, 15.

Solution

  1. 1The data are already in order.
  2. 2The two middle values are 8 and 10.
  3. 3(8 + 10) ÷ 2 = 9.

Answer: 9

Practice exercises 22.1

  1. Find the mean, median, mode, and range of 7, 9, 9, 10, 15.
  2. Find the median of 21, 4, 17, 9, 30, 12.
  3. Your first four test scores are 80, 92, 75, and 88. What score do you need on the fifth test for a mean of 85?
  4. Hourly pay at a small shop is $12, $12, $13, $14, and $49 (the owner). Which average describes a typical worker? Explain.
  5. A bowler's mean over three games is 150. Two games were 140 and 165. What was the third?
  6. List a data set of five numbers that has no mode.

Recap quiz 22.1

1. What is the median of 2, 9, 4?
2. What is the mode of 5, 5, 6, 7, 7, 7?
3. Which measure is pulled most by one huge outlier?

Section 22.1 recap

  • Mean = sum ÷ count; median = middle of sorted data; mode = most frequent; range = max − min.
  • When an outlier is present, report the median as the typical value.
Answer key with explanations: Section 22.1

Practice exercises

  1. Sum 50, so mean 10; median 9; mode 9; range 15 − 7 = 8.
  2. Order: 4, 9, 12, 17, 21, 30. Middle two are 12 and 17, so median = 14.5.
  3. Needed total 5 × 85 = 425. Current total 335. 425 − 335 = 90.
  4. Mean = 100 ÷ 5 = $20; median = $13. The median, because the owner's $49 is an outlier.
  5. Total 3 × 150 = 450; 450 − 140 − 165 = 145.
  6. Any five different numbers, for example 2, 4, 6, 8, 10.

Recap quiz

  1. 4: Ordered: 2, 4, 9. The middle value is 4.
  2. 7: 7 appears three times, more than any other value.
  3. Mean: Every value is added into the mean, so one extreme value moves it.
22.2

Probability: Measuring Chance

Learning objectives

After this section you will be able to:

  1. Find theoretical probability by counting equally likely outcomes.
  2. Find experimental probability from game stats.
  3. Use complements and the multiplication rule for independent events.

Real-life situation

Free throws in the final minute

A forward has made 42 of his 60 free throws this season. His team is down by one with seconds left, and he is fouled for two shots. Commentators say he is "a 70% shooter." What does that number really tell us?

AttemptsMadeMissedFT %
60421870.0%
Figure 22.2 Season free-throw stats (hypothetical player)

The concept

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, a decimal, or a percent.

Theoretical probability comes from counting: if every outcome is equally likely, P(event) = favorable outcomes ÷ total outcomes. Experimental probability comes from what actually happened: successes ÷ trials. Sports percentages are experimental probabilities.

The complement of an event is everything that is not the event, and P(not A) = 1 − P(A). Two events are independent when one does not change the chance of the other. For independent events, P(A and B) = P(A) × P(B).

Probability describes the long run. A 70% shooter will still miss sometimes, and a streak of makes does not mean a miss is "due."

Probability: equally likely outcomes

1
2
3
4
5
6
  1. An illustrative fair six-sided die has six equally likely outcomes.
  2. Three outcomes are even: 2, 4, 6.
  3. P(even) = 3 ÷ 6 = 1/2 = 50%. Equal likelihood is an assumption, not a rule for every real-world event.

Original WealthWiseGuys diagram · Hypothetical teaching quantities.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • probability

    The chance an event happens, written as a fraction, decimal, or percent between 0 and 1.

  • odds

    Another way to write probability, comparing the chance of an event to the chance it does not happen.

  • expected value

    The average result you would expect if a random situation were repeated many times.

Key terms

Outcome
One possible result of an experiment, like rolling a 4.
Event
A set of outcomes you are interested in, like rolling an even number.
Theoretical probability
Favorable outcomes ÷ total equally likely outcomes.
Experimental probability
Number of successes ÷ number of trials.
Complement
The event not happening.
Independent events
Events where one result does not affect the other.

Formulas & rules

  • Probability

    P(A) = favorable outcomes ÷ total outcomes

    0 ≤ P(A) ≤ 1

  • Complement

    P(not A) = 1 − P(A)

  • Independent events

    P(A and B) = P(A) × P(B)

Worked examples

Example 1Drawing from a bag

A bag holds 3 red, 5 blue, and 2 green tokens. One is drawn at random. Find P(blue) and P(not green).

Solution

  1. 1Total tokens: 3 + 5 + 2 = 10.
  2. 2P(blue) = 5 ÷ 10 = 1/2 = 0.5.
  3. 3P(green) = 2 ÷ 10 = 1/5, so P(not green) = 1 − 1/5 = 4/5 = 0.8.

Answer: P(blue) = 1/2; P(not green) = 4/5

Example 2Two free throws

Using 42 makes in 60 attempts, find the chance he makes both shots and the chance he misses both. Assume the shots are independent.

Solution

  1. 1Experimental probability of a make: 42 ÷ 60 = 0.7.
  2. 2P(miss) = 1 − 0.7 = 0.3.
  3. 3P(make both) = 0.7 × 0.7 = 0.49.
  4. 4P(miss both) = 0.3 × 0.3 = 0.09.
  5. 5Key point: the chance of at least one make is 1 − 0.09 = 0.91.

Answer: Make both 49%; miss both 9%

Example 3Rolling two dice

Two fair dice are rolled. What is the probability that the sum is 7?

Solution

  1. 1Each die has 6 faces, so there are 6 × 6 = 36 equally likely outcomes.
  2. 2Pairs that sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). That is 6 outcomes.
  3. 3P(sum 7) = 6 ÷ 36 = 1/6.

Answer: 1/6, about 16.7%

Practice exercises 22.2

  1. A fair coin is flipped twice. What is P(heads both times)?
  2. A spinner has 8 equal sections; 3 are gold. Find P(gold) as a fraction and a percent.
  3. A batter has 27 hits in 90 at-bats. What is his batting average (experimental probability of a hit)?
  4. The forecast gives a 35% chance of rain. What is the chance it does not rain?
  5. One card is drawn from a standard 52-card deck. Find P(heart) and P(not a heart).
  6. A player has won 5 coin flips in a row. What is the probability the next flip is heads? Explain.

Recap quiz 22.2

1. Every probability is between:
2. If P(A) = 0.2, then P(not A) is:
3. Rolling one fair die, P(5) is:

Section 22.2 recap

  • P(A) = favorable ÷ total; experimental P = successes ÷ trials.
  • P(not A) = 1 − P(A). For independent events, multiply.
  • Past results do not make an independent outcome "due."
Answer key with explanations: Section 22.2

Practice exercises

  1. 1/2 × 1/2 = 1/4.
  2. 3/8 = 37.5%.
  3. 27 ÷ 90 = 0.300.
  4. 1 − 0.35 = 0.65, or 65%.
  5. 13/52 = 1/4; complement 3/4.
  6. 1/2. Flips are independent; past results do not change the coin.

Recap quiz

  1. 0 and 1: 0 means impossible and 1 means certain (0% to 100%).
  2. 0.8: 1 − 0.2 = 0.8.
  3. 1/6: One favorable face out of six.
22.3

Odds, Expected Value, and Risk Literacy

Learning objectives

After this section you will be able to:

  1. Convert American odds to implied probability.
  2. Explain why the implied probabilities on an odds board add to more than 100%.
  3. Use expected value to see the long-run cost of a bet or game of chance.

Real-life situation

The odds on the broadcast

Sports broadcasts and apps now show odds on screen during games. You will see them whether or not you ever bet. This lesson teaches you to read them as a math and risk problem. In Pennsylvania you must be 21 to place a sports bet, and this lesson is not advice to gamble.

Hypothetical Championship · Team A vs Team B

Team A-200implied 66.7%
Team B+170implied 37.0%

Hypothetical odds for a math lesson. Sports betting is for adults 21+ in Pennsylvania.

Figure 22.3 Odds board for a hypothetical championship game

The concept

American odds are written with a plus or minus sign. A plus number such as +170 shows the profit on a $100 winning bet. A minus number such as −200 shows how much you must risk to profit $100. The favorite has the minus sign.

Odds can be turned into an implied probability. For positive odds: 100 ÷ (odds + 100). For negative odds: |odds| ÷ (|odds| + 100). If the two sides of a fair game were priced honestly, the probabilities would add to exactly 100%.

On a real board they add to more than 100%. The amount above 100% is called the vig or overround. It measures the book's pricing spread — it is not a guaranteed profit margin, and it does not tell you any single bettor's expected loss; those depend on the true probabilities and how the money is spread.

Expected value (EV) is the average result per play if you repeated a choice many times: multiply each result by its probability and add. A negative EV means that, over time, you should expect to lose money. Nearly every casino game, lottery, and sportsbook line is built to have negative EV for the player.

Risk literacy means knowing the math before someone else uses it on you. If gambling stops being fun or feels out of control for you or someone you know, the free, confidential helpline is 1-800-GAMBLER.

Odds, Expected Value, and Risk Literacy: solution map

  1. 1

    Identify the question

    Find the implied probability for Team A at −200 and Team B at +170.

  2. 2

    Calculation 1

    Team A (negative): 200 ÷ (200 + 100) = 200 ÷ 300 ≈ 0.6667, about 66.7%.

  3. 3

    Calculation 2

    Team B (positive): 100 ÷ (170 + 100) = 100 ÷ 270 ≈ 0.3704, about 37.0%.

  4. 4

    Calculation 3

    Total: 66.7% + 37.0% = 103.7%.

  5. 5

    Calculation 4

    Key point: the extra is about 3.7 percentage points of overround. A fair board would total 100%.

  6. 6

    Interpret and check

    Team A ≈ 66.7%, Team B ≈ 37.0%, total ≈ 103.7%

Figure 22.3.V Odds, Expected Value, and Risk Literacy: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • speculation

    A very risky investment made in hopes of a quick, large gain.

  • inflation

    A general rise in prices over time, which makes money buy less.

  • yield

    The income an investment pays, shown as a percent of its price.

Key terms

Odds
A way of stating the payout of a bet, which also implies a probability.
Favorite / underdog
The side expected to win (minus odds) / expected to lose (plus odds).
Implied probability
The chance of winning that the odds suggest.
Vig (overround)
How far implied probabilities exceed 100% — the book's pricing spread, not a guaranteed profit margin.
Expected value
Average gain or loss per play over many plays.

Formulas & rules

  • Positive odds

    Implied P = 100 ÷ (odds + 100)

  • Negative odds

    Implied P = |odds| ÷ (|odds| + 100)

  • Expected value

    EV = P(win) × gain − P(lose) × loss

Worked examples

Example 1Reading the board

Find the implied probability for Team A at −200 and Team B at +170.

Solution

  1. 1Team A (negative): 200 ÷ (200 + 100) = 200 ÷ 300 ≈ 0.6667, about 66.7%.
  2. 2Team B (positive): 100 ÷ (170 + 100) = 100 ÷ 270 ≈ 0.3704, about 37.0%.
  3. 3Total: 66.7% + 37.0% = 103.7%.
  4. 4Key point: the extra is about 3.7 percentage points of overround. A fair board would total 100%.

Answer: Team A ≈ 66.7%, Team B ≈ 37.0%, total ≈ 103.7%

Example 2The cost of a "50/50" bet

Both sides of an even game are listed at −110. You risk $110 to win $100. If the game really is a coin flip, what is the expected value of one bet?

Solution

  1. 1P(win) = 0.5, gain = $100. P(lose) = 0.5, loss = $110.
  2. 2EV = 0.5 × 100 − 0.5 × 110 = 50 − 55 = −$5.
  3. 3Over 100 such bets, the expected result is 100 × (−5) = −$500.

Answer: EV = −$5 per bet

Example 3A raffle

A school raffle sells 500 tickets at $2 each for one $400 prize. What is the expected value of one ticket?

Solution

  1. 1P(win) = 1 ÷ 500. Expected prize = 400 × 1/500 = $0.80.
  2. 2Subtract the ticket cost: 0.80 − 2.00 = −$1.20.
  3. 3Key point: a fundraiser raffle is honest about this, since the extra money supports the school. Commercial gambling keeps it.

Answer: EV = −$1.20 per ticket

Practice exercises 22.3

  1. Convert +300 to an implied probability.
  2. Convert −150 to an implied probability.
  3. At +250, what is the profit on a winning $20 bet?
  4. At −150, what is the profit on a winning $30 bet?
  5. A game costs $1 to play. You win $4 profit with probability 1/6; otherwise you lose the $1. Find the EV.
  6. A board lists +100 and +100 on the two sides. What do the implied probabilities total, and is there a margin?

Recap quiz 22.3

1. +100 odds imply a probability of:
2. Implied probabilities on a board total 104%. The extra 4 percentage points is:
3. A negative expected value means:

Section 22.3 recap

  • Plus odds = profit on $100; minus odds = risk needed to profit $100.
  • Implied probabilities over 100% reveal the overround — the book's pricing spread, not a guaranteed profit.
  • Negative EV means the game is designed for you to lose over time.
Answer key with explanations: Section 22.3

Practice exercises

  1. 100 ÷ 400 = 0.25 = 25%.
  2. 150 ÷ 250 = 0.6 = 60%.
  3. 20 × 250 ÷ 100 = $50 profit.
  4. 30 × 100 ÷ 150 = $20 profit.
  5. (1/6)(4) − (5/6)(1) = 4/6 − 5/6 = −1/6 ≈ −$0.17 per play.
  6. 50% + 50% = 100%. No margin; real boards almost never do this.

Recap quiz

  1. 50%: 100 ÷ (100 + 100) = 0.5.
  2. The overround: The overround builds an advantage into the quoted payouts; it does not guarantee profit on every event.
  3. You expect to lose money over many plays: EV is a long-run average, not a prediction of one play.
22.4

Investing Returns and Misleading Charts

Learning objectives

After this section you will be able to:

  1. Compute percent change for a price or investment.
  2. Show why a gain and an equal-percent loss do not cancel.
  3. Spot a truncated axis and other misleading chart choices.

Real-life situation

"Resale prices are exploding!"

A post shows the resale price of a hypothetical sneaker over three months with towering bars. The caption says prices are "exploding." The same numbers drawn on a chart that starts at $0 tell a calmer story.

$178$190$180Jan$184Feb$190Mar
Figure 22.4a The post's chart: the axis starts at $178

The concept

Percent change compares the change to where you started: (new − old) ÷ old × 100%. It is how investment apps report gains and losses.

Percent gains and losses do not cancel. If an investment rises 50% and then falls 50%, the fall is 50% of a bigger number, so you end up below where you began. This is why steady growth beats wild swings.

Compound growth means each year's gain is earned on the new, larger balance: Balance = Start × (1 + r)ⁿ. Historical averages are not promises; real returns vary and can be negative.

Charts can mislead without lying about a single number. Watch for a vertical axis that does not start at zero, uneven time intervals, missing labels, cherry-picked start dates, and 3-D effects. Always read the axis before the bars.

$0$190$180Jan$184Feb$190Mar
Figure 22.4b The same data with the axis starting at $0

Investing Returns and Misleading Charts: solution map

  1. 1

    Identify the question

    The sneaker went from $180 in January to $190 in March. Find the percent change.

  2. 2

    Calculation 1

    Change: 190 − 180 = $10.

  3. 3

    Calculation 2

    Percent change: 10 ÷ 180 ≈ 0.0556 = 5.56%.

  4. 4

    Calculation 3

    Key point: in Figure 22.4a the March bar looks about 6 times taller than January, because the bars show only the part above $178 (12 vs. 2). The real increase is about 5.6%.

  5. 5

    Interpret and check

    About a 5.6% increase

Figure 22.4.V Investing Returns and Misleading Charts: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • return

    The money or gain an investment produces, often shown as a percent.

  • ROI

    Return on investment — the gain compared to what you put in, as a percent: (gain ÷ cost) × 100.

  • capital gain

    Profit made by selling an investment for more than it cost.

Key terms

Percent change
The change in a value as a percent of its starting value.
Return
The gain or loss on an investment, usually shown as a percent.
Compound growth
Growth where each period's gain is added before the next gain is figured.
Truncated axis
An axis that starts above zero, which exaggerates differences.

Formulas & rules

  • Percent change

    % change = (new − old) ÷ old × 100%

  • Compound growth

    A = P(1 + r)ⁿ

    P = start, r = rate per period as a decimal, n = periods.

Worked examples

Example 1How big is the jump?

The sneaker went from $180 in January to $190 in March. Find the percent change.

Solution

  1. 1Change: 190 − 180 = $10.
  2. 2Percent change: 10 ÷ 180 ≈ 0.0556 = 5.56%.
  3. 3Key point: in Figure 22.4a the March bar looks about 6 times taller than January, because the bars show only the part above $178 (12 vs. 2). The real increase is about 5.6%.

Answer: About a 5.6% increase

Example 2Up 50%, down 50%

You invest $1,000. It gains 50% one year and loses 50% the next. What do you have?

Solution

  1. 1Year 1: 1,000 × 1.50 = $1,500.
  2. 2Year 2: 1,500 × 0.50 = $750.
  3. 3Overall: (750 − 1,000) ÷ 1,000 = −25%.

Answer: $750, a 25% loss, even though the two percents "average" to 0%

Example 3Compound growth (hypothetical)

A student puts $500 in a broad index fund. Suppose it grows 7% per year for 2 years. Find the balance. (7% is a hypothetical rate, not a prediction.)

Solution

  1. 1After year 1: 500 × 1.07 = $535.00.
  2. 2After year 2: 535 × 1.07 = $572.45.
  3. 3Check with the formula: 500 × 1.07² = 500 × 1.1449 = $572.45. ✓

Answer: $572.45

Practice exercises 22.4

  1. A stock goes from $40 to $46. Find the percent change.
  2. A stock goes from $80 to $60. Find the percent change.
  3. $200 rises 10%, then falls 10%. What is the final value?
  4. A chart's axis starts at 95. Bars show 96 and 100. How many times taller does the second bar look, and what is the real percent increase?
  5. $1,000 grows 5% per year for 3 years. Find the balance to the nearest cent.
  6. Name two ways a chart can mislead besides a truncated axis.

Recap quiz 22.4

1. The most common trick in a misleading bar chart is:
2. $100 gains 20% and then loses 20%. The result is:
3. Percent change is found by dividing the change by the:

Section 22.4 recap

  • % change = (new − old) ÷ old. Gains and equal-percent losses do not cancel.
  • Compound growth: A = P(1 + r)ⁿ. Past returns are not promises.
  • Read the axis first. A chart that doesn't start at zero can make 5% look like 500%.
Answer key with explanations: Section 22.4

Practice exercises

  1. 6 ÷ 40 = 0.15 = +15%.
  2. −20 ÷ 80 = −0.25 = −25%.
  3. 200 × 1.10 = 220; 220 × 0.90 = $198.
  4. Visible heights 1 and 5, so it looks 5 times taller. Real increase 4 ÷ 96 ≈ 4.17%.
  5. 1,000 × 1.05³ = 1,000 × 1.157625 = $1,157.63.
  6. Any two: uneven time intervals, cherry-picked start or end dates, missing labels or units, 3-D distortion, different scales on side-by-side charts.

Recap quiz

  1. An axis that doesn't start at zero: A truncated axis exaggerates small differences.
  2. $96: 100 × 1.2 = 120; 120 × 0.8 = 96.
  3. Old value: Percent change compares to the starting (old) value.
22.5

Interpreting Business Data: Weighted Averages and Percent of Total

Learning objectives

After this section you will be able to:

  1. Calculate sales revenue and shares of a weekly total.
  2. Find a weighted average selling price using units as weights.
  3. Distinguish unit share, revenue share, and profit.

Real-life situation

Interpreting Business Data: Weighted Averages and Percent of Total in practice

A hypothetical Philadelphia food truck records one week of item sales below. Every item in a category sells at the listed price, with no discounts, refunds, or sales tax included. All amounts are invented; round final money to cents and percentages to two decimals.

ItemUnits soldPrice per unit ($)Revenue ($)
Rice bowl1208960
Wrap806480
Lemon drink2003600
Total400Varies2040
Figure 22.5 Interpreting Business Data: Weighted Averages and Percent of Total (hypothetical data)

The concept

A business table can describe several different totals. Units sold count individual items, while revenue counts dollars received from selling them. Multiply units by price for each category before adding revenue. A row containing a total is a summary, not an additional category to add again.

A percent of total compares one category with the matching grand total. The rice bowl unit share uses 120 divided by 400 items; its revenue share uses $960 divided by $2,040. Both are legitimate but answer different questions. Always name the denominator and the time period.

A weighted average gives more influence to values represented more often. To find average revenue per item sold, weight each listed price by its unit count: add price times units, then divide by total units. The ordinary mean of the three menu prices treats each category as equally common, which is not how this week’s sales occurred.

Average selling price is not average spending per customer because one customer may buy several items. Revenue share also is not profit share; ingredient and operating costs are absent. Rounded percentage shares may not total exactly 100%, so keep unrounded values until the final display and explain small rounding differences.

Interpreting Business Data: Weighted Averages and Percent of Total: solution map

  1. 1

    Identify the question

    Use the table to find total revenue and average revenue per item sold.

  2. 2

    Calculation 1

    Rice bowls: 120 × $8 = $960; wraps: 80 × $6 = $480.

  3. 3

    Calculation 2

    Drinks: 200 × $3 = $600.

  4. 4

    Calculation 3

    Total revenue = 960 + 480 + 600 = $2,040; total units = 120 + 80 + 200 = 400.

  5. 5

    Calculation 4

    Weighted average price = 2,040 ÷ 400 = $5.10.

  6. 6

    Interpret and check

    Total revenue is $2,040; the truck earns $5.10 per item on average, not per customer.

Figure 22.5.V Interpreting Business Data: Weighted Averages and Percent of Total: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • revenue

    The total money a business brings in from sales, before any costs are paid.

  • profit margin

    Profit shown as a percent of revenue; a 10% margin means $10 of profit per $100 of sales.

  • portfolio

    The full collection of investments a person or fund owns.

Key terms

Weight
The count or importance attached to a value in an average.
Weighted average
Sum of value times weight, divided by total weight.
Revenue
Selling price multiplied by quantity sold, summed across sales.
Percent of total
A category amount divided by the matching overall amount, times 100%.
Sales mix
The distribution of sales among product categories.

Formulas & rules

  • Item revenue

    Revenue = units sold × price per unit

  • Weighted average price

    Average price = sum(price × units) ÷ sum(units)

  • Percent of total

    Share = category amount ÷ matching total × 100%

Worked examples

Example 1Weekly average price

Use the table to find total revenue and average revenue per item sold.

Solution

  1. 1Rice bowls: 120 × $8 = $960; wraps: 80 × $6 = $480.
  2. 2Drinks: 200 × $3 = $600.
  3. 3Total revenue = 960 + 480 + 600 = $2,040; total units = 120 + 80 + 200 = 400.
  4. 4Weighted average price = 2,040 ÷ 400 = $5.10.

Answer: Total revenue is $2,040; the truck earns $5.10 per item on average, not per customer.

Example 2Two different shares

Find rice bowls’ unit share and revenue share.

Solution

  1. 1Identify unit denominator: 400 items.
  2. 2Unit share = 120 ÷ 400 × 100% = 30.00%.
  3. 3Identify revenue denominator: $2,040.
  4. 4Revenue share = 960 ÷ 2,040 × 100% ≈ 47.06%.

Answer: Rice bowls are 30.00% of units but 47.06% of revenue.

Practice exercises 22.5

  1. Find wraps’ unit share and revenue share.
  2. Find drinks’ revenue share.
  3. Find the simple mean of the three prices and explain why it is not this week’s average selling price.
  4. Another week sells 10 bowls at $8 and 30 drinks at $3. Find average price.
  5. A reader calls $2,040 the weekly profit. Explain what is missing.
  6. What do this week’s displayed revenue shares sum to?

Recap quiz 22.5

1. Correct denominator for drinks’ unit share?
2. Average selling price for this week?
3. Does the table reveal profit?

Section 22.5 recap

  • Weight prices by units sold, not by the number of menu categories.
  • Name the denominator: unit share and revenue share answer different questions.
  • Revenue and average selling price do not reveal profit or spending per customer.
Answer key with explanations: Section 22.5

Practice exercises

  1. Unit share = 80 ÷ 400 × 100% = 20.00%; revenue share = 480 ÷ 2,040 × 100% ≈ 23.53%.
  2. Drink revenue = 200 × 3 = $600; 600 ÷ 2,040 × 100% ≈ 29.41%.
  3. (8 + 6 + 3) ÷ 3 = 17 ÷ 3 ≈ $5.67. It weights categories equally, unlike the actual 120, 80, and 200 unit counts; the weighted result is $5.10.
  4. Revenue = 10 × 8 + 30 × 3 = $170; units = 40; weighted average = 170 ÷ 40 = $4.25.
  5. The $2,040 equals sales revenue, 960 + 480 + 600. Profit requires subtracting food, labor, and other costs; those costs are not supplied.
  6. 47.06% + 23.53% + 29.41% = 100.00%. Other data sets may differ slightly from 100% after rounding, even when the exact shares total 100%.

Recap quiz

  1. 400 items: The category has 200 units out of 400 total units.
  2. $5.10: Total revenue $2,040 ÷ 400 units = $5.10.
  3. No, costs are missing: Revenue minus costs gives profit; the table has no cost data.

Chapter 22 Organizer and Review

Chapter Organizer

Every section at a glance: key formulas, key terms, and the main idea.

  • 22.1 Mean, Median, Mode, and Range

    Key formulas

    • Mean: Mean = (sum of values) ÷ (number of values)
    • Range: Range = maximum − minimum
    • Median (even count): Median = (middle value 1 + middle value 2) ÷ 2

    Key terms

    Data set, Mean, Median, Mode, Range, Outlier

    Remember

    Mean = sum ÷ count; median = middle of sorted data; mode = most frequent; range = max − min.

  • 22.2 Probability: Measuring Chance

    Key formulas

    • Probability: P(A) = favorable outcomes ÷ total outcomes
    • Complement: P(not A) = 1 − P(A)
    • Independent events: P(A and B) = P(A) × P(B)

    Key terms

    Outcome, Event, Theoretical probability, Experimental probability, Complement, Independent events

    Remember

    P(A) = favorable ÷ total; experimental P = successes ÷ trials.

  • 22.3 Odds, Expected Value, and Risk Literacy

    Key formulas

    • Positive odds: Implied P = 100 ÷ (odds + 100)
    • Negative odds: Implied P = |odds| ÷ (|odds| + 100)
    • Expected value: EV = P(win) × gain − P(lose) × loss

    Key terms

    Odds, Favorite / underdog, Implied probability, Vig (overround), Expected value

    Remember

    Plus odds = profit on $100; minus odds = risk needed to profit $100.

  • 22.4 Investing Returns and Misleading Charts

    Key formulas

    • Percent change: % change = (new − old) ÷ old × 100%
    • Compound growth: A = P(1 + r)ⁿ

    Key terms

    Percent change, Return, Compound growth, Truncated axis

    Remember

    % change = (new − old) ÷ old. Gains and equal-percent losses do not cancel.

  • 22.5 Interpreting Business Data: Weighted Averages and Percent of Total

    Key formulas

    • Item revenue: Revenue = units sold × price per unit
    • Weighted average price: Average price = sum(price × units) ÷ sum(units)
    • Percent of total: Share = category amount ÷ matching total × 100%

    Key terms

    Weight, Weighted average, Revenue, Percent of total, Sales mix

    Remember

    Weight prices by units sold, not by the number of menu categories.

Cumulative Review

Mixed practice from this chapter and the two chapters before it.

  1. Section 22.1

    List a data set of five numbers that has no mode.

    Show answer

    Any five different numbers, for example 2, 4, 6, 8, 10.

  2. Section 22.2

    A player has won 5 coin flips in a row. What is the probability the next flip is heads? Explain.

    Show answer

    1/2. Flips are independent; past results do not change the coin.

  3. Section 22.3

    A board lists +100 and +100 on the two sides. What do the implied probabilities total, and is there a margin?

    Show answer

    50% + 50% = 100%. No margin; real boards almost never do this.

  4. Section 22.4

    Name two ways a chart can mislead besides a truncated axis.

    Show answer

    Any two: uneven time intervals, cherry-picked start or end dates, missing labels or units, 3-D distortion, different scales on side-by-side charts.

  5. Section 22.5

    What do this week’s displayed revenue shares sum to?

    Show answer

    47.06% + 23.53% + 29.41% = 100.00%. Other data sets may differ slightly from 100% after rounding, even when the exact shares total 100%.

  6. Review of Section 20.3

    A 2% chance of a $5,000 loss otherwise means no loss. Find expected loss.

    Show answer

    0.02 × $5,000 + 0.98 × $0 = $100.00.

  7. Review of Section 21.5

    Go or No-Go? You buy 4 shares at $10.00 and sell at $10.50. The spread costs a combined $0.10 per share and the app charges a $2 fee on the sale. Find the net result and explain your decision.

    Show answer

    Gross 4 × $0.50 = $2. Costs 4 × $0.10 = $0.40 plus $2 = $2.40. Net $2 − $2.40 = −$0.40. A gain on the screen became a loss after costs; No-Go at this size.

All prices, stores, people, and accounts in this chapter are hypothetical and for teaching only. Product references are generic categories; no brand endorses or is endorsed by this textbook.

Chapter 21Chapter 23