HSS.MD.A.4Common CoreMathStatistics and ProbabilityGrades 9-12
HSS.MD.A.4: Probability Distributions from Data and Expected Value
In plain English: HSS.MD.A.4 is an advanced (+) Common Core statistics and probability standard that asks students to build a probability distribution for a random variable from observed data, using relative frequencies as the probabilities, and then find its expected value. It is usually taught in Precalculus, Algebra II or a statistics course.
(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
Common Core State Standards for Mathematics · Domain: Using Probability to Make Decisions (MD) · Cluster: Calculate expected values and use them to solve problems Also written as HSS-MD.A.4 or S-MD.4 · Official standard
Students learn to build a probability distribution when no formula or equally likely model is available: they define a random variable, collect or read data, turn each frequency into a relative frequency and use those relative frequencies as probabilities. They then find the expected value, E(X) = Σ x · P(X = x), and use it to predict totals for many trials, such as the number of TV sets in 100 households.
The standard is the data-based partner of HSS.MD.A.3, where probabilities come from theory. Here the probabilities are estimates, so the lesson also asks how the sample, its size and open-ended categories such as "4 or more" affect the distribution and the expected value.
Learning Objectives
By the end of this lesson, students will be able to:
Define a random variable for a real situation and list its possible values
Develop an empirical probability distribution from a frequency table by computing relative frequencies, and check that it is valid
Find the expected value of a random variable from an empirical distribution and interpret it as a long-run average
Use the expected value to predict a total for a given number of trials, such as the number of TV sets in 100 households
Explain how the sample and the way categories are recorded affect an empirical distribution
Prior Knowledge Required
Students should already be comfortable with:
Approximating a probability by a long-run relative frequency 7.SP.C.6
Probability models built from observed frequencies, which may not be uniform 7.SP.C.7
The mean of a data set as a measure of center 6.SP.B.5
Random variables and expected value as the mean of a probability distribution HSS.MD.A.2
Project a small frequency table from an invented survey of 50 households and ask students to work alone for three minutes, then compare with a partner.
Cars per household in 50 households (invented data, warm-up)
Cars in household
Number of households
0
4
1
17
2
20
3
9
Total
50
Warm-Up Prompt
"If you pick one of these 50 households at random, what is the chance it has exactly 2 cars? At least 2 cars? What is the average number of cars per household?"
Collect answers: exactly 2 cars is 20/50 = 0.40, and at least 2 cars is (20 + 9)/50 = 0.58. For the average, some students add 0 + 1 + 2 + 3 and divide by 4. Ask them why that ignores the data, then compute (0 · 4 + 1 · 17 + 2 · 20 + 3 · 9)/50 = 84/50 = 1.68. Tell students that the fractions they found are the probabilities of an empirical distribution, and 1.68 is its expected value.
Direct Instruction20-25 minutes
Explain the difference between the two ways to assign probabilities. In HSS.MD.A.3, probabilities come from a theoretical model (a fair die, a shuffled deck). In HSS.MD.A.4, no such model exists (how many TV sets does a household own?), so probabilities are assigned empirically: they are relative frequencies from data. Then give the procedure:
Define the random variable in words, with units: "X = the number of TV sets in a randomly selected household."
Collect or read the data and tally how often each value of X occurs. Decide how to record open-ended groups such as "4 or more".
Turn frequencies into probabilities: P(X = x) = (frequency of x) / (total number of observations).
Check the distribution: every probability is between 0 and 1 and the probabilities add to 1 (up to rounding).
Find the expected value: E(X) = Σ x · P(X = x). It equals the mean of the data, and it predicts the long-run average per trial.
Predict totals: for n new trials, expect about n · E(X), and say why the prediction is only approximate.
Work through the examples below. For each one, have students say what the random variable is before any computing.
Official example: TV sets per household
An illustrative table (not official data) for 500 households: 0 TVs: 25, 1 TV: 140, 2 TVs: 165, 3 TVs: 110, 4 or more: 60. Build the distribution, find the expected number of sets per household, and predict the number of sets in 100 randomly selected households.
After Example 1, show Diagram 1. The expected value 2.08 is the balance point of the bar graph. Because households with 5 or more sets were recorded as 4, the true mean is probably a little higher than 2.08: ask students which direction the error goes and why. After Example 3, stress that an expected value does not have to be a possible value of X: no day has 1.95 orders, but over many days the average is close to 1.95.
Guided Practice15 minutes
Pairs work from the table below, one question at a time, and the class compares answers after each step. (a) Define X. (b) Build the probability distribution. (c) Find P(X ≥ 3). (d) Find E(X). (e) Predict the number of people in the next 1,000 cars.
People per car for 200 cars at a gas station (invented data, guided practice)
People in car
1
2
3
4
5
Number of cars
96
62
26
12
4
Answers: X = number of people in a randomly selected car; P = 0.48, 0.31, 0.13, 0.06, 0.02, which add to 1; P(X ≥ 3) = 0.21; E(X) = 0.48 + 0.62 + 0.39 + 0.24 + 0.10 = 1.83 people per car; about 1,830 people in 1,000 cars. Listen for pairs that divide by 5 (the number of categories) instead of 200, and for pairs that multiply the counts by the values but forget to divide by the total.
Independent Practice15 minutes
Students complete three problems on their own:
A library counts books checked out per visit for 80 visits (invented data): 0 books: 12, 1: 28, 2: 24, 3: 16. Build the distribution and find E(X). (P = 0.15, 0.35, 0.30, 0.20; E(X) = 1.55 books.)
A distribution for X = 0, 1, 2, 3 has P(X = 0) = 0.10, P(X = 1) = 0.40 and P(X = 2) = 0.30. Find P(X = 3) and E(X). (P(X = 3) = 0.20; E(X) = 1.6.)
A streaming company estimates the distribution of screens per household by surveying only people who already subscribe. In two sentences, explain how this could bias the distribution and the expected value.
Closure5 minutes
Exit ticket: A coffee shop records free refills for 100 customers (invented data): 0 refills: 45 customers, 1 refill: 35, 2 refills: 20. (1) Write the probability distribution. (2) Find the expected number of refills per customer. (3) How many refills should the shop plan for 400 customers? (Answers: 0.45, 0.35, 0.20; E(X) = 0.75; about 300 refills.)
Differentiation Strategies
For Struggling Students
Give a three-column table template with the headings x, P(X = x) and x · P(X = x), so the expected value is the sum of the last column
Start with data sets whose total is 100, so each relative frequency is the count written as a decimal
Have students write the random variable as a sentence ("X = the number of ...") before touching any numbers
For Advanced Students
Ask students to compute the standard deviation of an empirical distribution and describe what it adds to the expected value
Have students split one large data set into random halves, build a distribution from each half and explain why the two expected values differ
Ask how a survey that reports "4 or more" could be used to find a lower bound for the true mean, and what extra information would give an upper bound
Assessment Guidance
What to Look For
Check that students divide every frequency by the total number of observations and verify that the probabilities add to 1 before they compute E(X). When students find the expected value, look for a weighted sum, not the average of the listed values. In interpretations, listen for "average per household over many households" rather than "each household has". For predictions, students should multiply the expected value by the number of trials and say that the result is approximate because the probabilities came from a sample.
02
Classroom Activities
3 Activities
1
Bottle Flip Distribution
20 minGroups of 3-4
Students generate their own data for a random variable that has no theoretical model: X = the number of successful bottle flips (the bottle lands upright) in a round of 3 flips. The class pools its rounds, builds an empirical distribution and finds E(X).
Procedure
Each group gets one plastic water bottle filled about one third with water. Agree as a class on the flipping rules (same height, one hand, land on a desk)
A round is 3 flips by the same student. Record X, the number of flips that land upright (0, 1, 2 or 3). Each group completes 15 rounds, rotating flippers
Groups build their own relative frequency table and find their own E(X)
Pool all groups' rounds on the board, build the class distribution and compute the class E(X)
Discussion Questions
Why can we not find these probabilities by assuming each outcome is equally likely?
Which do you trust more, your group's E(X) or the class E(X)? Why?
Based on the class distribution, how many successful flips would you expect in 40 rounds?
Would the distribution change if a different student did all the flipping? What does that say about the "sample space" we used?
Modification for Distance Learning
Students flip a paper cup at home (X = number of times out of 3 that it lands on its side) and enter their rounds in a shared spreadsheet. The teacher builds the class distribution live on screen.
2
From Our Class to the Whole School
20 minWhole class, then pairs
The class collects anonymous survey data on one household variable, builds an empirical distribution, and uses the expected value to predict a total for all the households in the school. Pairs then critique their own prediction.
Procedure
Choose a variable as a class, such as X = the number of pets in a student's household. Students write their value on a slip of paper without names
Tally the slips, compute relative frequencies and check that they add to 1
Pairs compute E(X) and predict the total number of pets in the households of all students in the school, using the school's enrollment
Each pair writes two reasons the prediction could be off
Discussion Questions
Is our class a random sample of the school? What kind of student might be missing?
Siblings at the same school share a household. How could that affect the prediction?
What would you change to make the estimate more trustworthy?
Research Variation
Work the official example with current data. Students look up a national survey table on the number of televisions per household, such as the U.S. Energy Information Administration's Residential Energy Consumption Survey (RECS). They build the distribution from the reported counts or percents, decide how to code the top category, compute the expected number of sets per household and predict the number of sets in 100 randomly selected households.
3
Spot the Error Gallery Walk
20 minPairs
Six stations around the room each show a student's work with one error. Pairs rotate every 3 minutes, name the error, and write the correct result on a sticky note.
The Six Stations
Station 1: A student lists P(X = 0) = 0.2, P(X = 1) = 0.5, P(X = 2) = 0.4 as a distribution. (Error: the probabilities add to 1.1, so this is not a distribution.)
Station 2: Counts of 5, 10 and 5 for X = 1, 2, 3 in 20 observations; the student reports E(X) = 1(5) + 2(10) + 3(5) = 40. (Error: used counts instead of probabilities; E(X) = 0.25 + 1.00 + 0.75 = 2.)
Station 3: Counts of 10, 20, 50 and 20 for X = 0, 1, 2, 3 in 100 observations; the student reports E(X) = (0 + 1 + 2 + 3)/4 = 1.5. (Error: ignored the frequencies; E(X) = 0.2 + 1.0 + 0.6 = 1.8.)
Station 4: A team scored 0, 1, 2 and 3 goals in 4, 6, 6 and 4 of 20 games. The student computes E(X) = 1.5 correctly and writes "the team will score 1.5 goals in the next game." (Error: interpretation; 1.5 is a long-run average per game.)
Station 5: E(X) = 2.4 items per order; the student predicts 2.4 + 250 = 252.4 items for 250 orders. (Error: should multiply; about 600 items.)
Station 6: Counts of 30, 45, 15 and 10 for X = 1, 2, 3, 4 in 100 observations; the student leaves out X = 4 and divides the others by 90. (Error: a value is missing; P = 0.30, 0.45, 0.15, 0.10 and E(X) = 2.05.)
Procedure
Print each station on a large sheet and post it with a blank area for sticky notes
At each station, one partner names the error and the other writes the corrected answer; they switch roles at the next station
After the walk, each pair writes a checklist of four things to verify before reporting an expected value
Challenge Variation
Pairs write their own "error station" from a new data set and trade with another pair, who must find and fix the error.
03
Diagrams & Visual Aids
1 diagrams
Diagram 1: An Empirical Distribution and Its Expected Value
Relative frequencies from an illustrative survey of 500 households, drawn to scale. Each bar is a probability assigned from data, and the bars add to 1. The expected value, 2.08 sets per household, is the balance point of the distribution. Households with 5 or more sets were recorded as 4, so the true mean is probably slightly higher.
04
Homework Assignment
~30 min
HSS.MD.A.4 Homework: Distributions from Data and Expected Value
Directions: For each problem, define the random variable in words, show the relative frequency for every value, and check that the probabilities add to 1. Show the sum used for each expected value, and write one sentence interpreting it in the context. All data sets are invented for practice unless the problem asks you to find real data.
Part 1: Building Empirical Distributions (Problems 1-3)
A dentist's office records the number of cavities found at 150 checkups: 0 cavities: 84 checkups, 1 cavity: 39, 2 cavities: 18, 3 cavities: 9. (a) Build the probability distribution of X = number of cavities at a checkup. (b) Show that it is a valid distribution. (c) Find the probability that a checkup finds at least one cavity.
An animal shelter records adoptions per day for 40 days: 0 adoptions: 8 days, 1: 14 days, 2: 10 days, 3: 6 days, 4: 2 days. (a) Build the probability distribution. (b) Find the expected number of adoptions per day. (c) How many adoptions should the shelter expect in a 30-day month?
A hotel records guests per room for 400 room-nights: 1 guest: 88, 2 guests: 172, 3 guests: some number, 4 guests: 40. (a) Find the missing count. (b) Build the probability distribution. (c) Find the expected number of guests per room-night.
Part 2: Expected Value, Predictions and Data Quality (Problems 4-6)
A food truck records extra toppings ordered on 500 tacos: 0 extras: 210 tacos, 1 extra: 190, 2 extras: 75, 3 extras: 25. Each extra topping costs the customer $0.50. (a) Find the expected number of extras per taco. (b) Predict the revenue from extras on 1,200 tacos.
Two groups estimate the distribution of X = bicycles per household. Group A surveys 40 households: 1 bike: 30%, 2 bikes: 50%, 3 bikes: 20%. Group B surveys 400 randomly selected households: 1 bike: 25%, 2 bikes: 45%, 3 bikes: 25%, 4 bikes: 5%. (a) Find E(X) for each group. (b) Which estimate would you use for the town, and why? Give two reasons.
Official example with current data: Find a current national data table on the number of TV sets per household (for example, from the U.S. Energy Information Administration's Residential Energy Consumption Survey). Record the source and year. (a) Build the probability distribution and explain how you coded the top category. (b) Find the expected number of sets per household. (c) How many TV sets would you expect in 100 randomly selected households? (d) Is your answer to (c) likely too high or too low because of the top category? Explain.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Building the Distribution
Random variable defined; every relative frequency correct; sum checked
Minor arithmetic error or sum not checked
Counts used as probabilities or values missing
Expected Value
Weighted sum shown and correct
Correct method with one arithmetic error
Unweighted average or no method shown
Prediction and Interpretation
Total predicted with n · E(X) and interpreted as an approximate long-run result
Correct number but interpretation missing or says every unit has E(X)
Missing or incorrect
Data Quality
Explains sample size, sampling method and category coding with specific reasons
One relevant reason given
No reasoning about the data
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again. All data sets are invented for practice.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A county health office wants a probability distribution for X = the number of emergency room visits a resident made last year. Which approach gives an empirical distribution?
Answer: B
An empirical distribution assigns each probability from observed data, as a relative frequency. Choices A and D assign theoretical, equally likely probabilities, which is the HSS.MD.A.3 approach and has no basis here. Choice C gives one estimate of the center, not a distribution.
Question 2 of 20 · Multiple Choice
In a survey of 80 students (invented data), the number of siblings was 0 for 12 students, 1 for 36, 2 for 20 and 3 for 12. What is P(X = 1)?
Answer: C
P(X = 1) = 36/80 = 0.45. Choice A divides by 100 as if the counts were percents. Choice B gives each of the four values the same probability, which ignores the data. Choice D is the count, not a probability.
Question 3 of 20 · Multiple Choice
An empirical distribution has P(X = 0) = 0.15, P(X = 1) = 0.25, P(X = 2) = 0.30 and P(X = 4) = 0.10. The only other value is X = 3. What is P(X = 3)?
Answer: A
The probabilities must add to 1: 0.15 + 0.25 + 0.30 + 0.10 = 0.80, so P(X = 3) = 1 - 0.80 = 0.20. Choice C is the sum of the known probabilities, the step before subtracting from 1.
Question 4 of 20 · Multiple Choice
Which list can be a probability distribution for X = 0, 1, 2?
Answer: D
In D every probability is between 0 and 1 and the sum is 1. Choice A adds to 1.2. Choice B adds to 1 but contains a negative probability. Choice C lists counts; dividing by their total, 100, would give a valid distribution.
Question 5 of 20 · Multiple Choice
X has P(X = 1) = 0.5, P(X = 2) = 0.3 and P(X = 3) = 0.2. What is E(X)?
Answer: B
E(X) = 1(0.5) + 2(0.3) + 3(0.2) = 0.5 + 0.6 + 0.6 = 1.7. Choice A is the unweighted average of the values 1, 2 and 3. Choice C divides the correct sum by 3, which is not part of the formula. Choice D adds the values.
Question 6 of 20 · Multiple Choice
Based on a year of records, the expected number of calls a fire station receives in an 8-hour shift is 3.4. Which interpretation is correct?
Answer: A
An expected value is a long-run average per trial. Choice C treats it as a prediction for a single shift, and a shift cannot have 3.4 calls. Choice B confuses the mean with the only possible values. Choice D reads the expected value as a percent.
Question 7 of 20 · Multiple Choice
An online store's data give an expected 0.35 returned items per order. About how many returned items should the store expect from 2,000 orders?
Answer: C
Multiply the expected value per order by the number of orders: 2,000 × 0.35 = 700. Choice D divides 2,000 by 0.35 instead of multiplying. Choice B is the expected value for one order.
Question 8 of 20 · Multiple Choice
Over 25 school days, the number of late buses was 0 on 10 days, 1 on 9 days and 2 on 6 days. What is P(X = 2) for X = the number of late buses on a day?
Answer: D
P(X = 2) = 6/25 = 0.24. Choice C gives each of the three values probability 1/3, which ignores the data. Choice B is the count. Choice A divides 6 by 100 instead of by 25.
Question 9 of 20 · Multiple Choice
Use the late-bus data from the previous question (0 late buses on 10 days, 1 on 9 days, 2 on 6 days out of 25). What is the expected number of late buses per day?
Answer: A
E(X) = (0 · 10 + 1 · 9 + 2 · 6)/25 = 21/25 = 0.84. Choice D is the total number of late buses over 25 days, not the average per day. Choice C divides that total by the 3 values instead of the 25 days. Choice B is the middle value.
Question 10 of 20 · Multiple Choice
A class of 15 students and a random sample of 300 students from the whole grade each build an empirical distribution of X = books read over the summer. Why is the distribution from 300 students a better estimate for the grade?
Answer: C
Larger random samples tend to give relative frequencies closer to the true proportions. Choice A overstates this: any sample still gives estimates. Choice B is false, because relative frequencies from any complete sample add to 1.
Question 11 of 20 · Multiple Choice
X = pets per household for 50 households: 0 pets: 20 households, 1 pet: 15, 2 pets: 10, 3 pets: 5. A student says E(X) = (0 + 1 + 2 + 3)/4 = 1.5. What is the error, and what is E(X)?
Answer: B
Each value must be weighted by its probability: E(X) = (0 · 20 + 1 · 15 + 2 · 10 + 3 · 5)/50 = 50/50 = 1. Choice C divides the sum of the values, not the weighted sum, by 50. Choice D is the number of households.
Question 12 of 20 · Multiple Choice
At a coffee kiosk, 70% of customers buy 1 cup, 25% buy 2 cups and 5% buy 3 cups. What is the expected number of cups per customer?
Answer: D
E(X) = 1(0.70) + 2(0.25) + 3(0.05) = 0.70 + 0.50 + 0.15 = 1.35. Choice A averages the values 1, 2 and 3 without weights. Choice B is the sum of the probabilities, which is always 1. Choice C divides the correct answer by 3.
Question 13 of 20 · Multiple Choice
An invented survey table of TV sets per household gives an expected value of 2.3 sets per household. How many TV sets would you expect in 100 randomly selected households?
Answer: C
The expected total is 100 × 2.3 = 230 sets. Choice A is the expected number for one household. Choice B multiplies by 10 instead of 100. Choice D assumes one set per household.
Question 14 of 20 · Multiple Choice
For which random variable must the probabilities be assigned empirically, from data, rather than calculated from equally likely outcomes?
Answer: A
Arrivals at a food truck have no equally likely model, so their probabilities must come from observed data. Choices B, C and D have theoretical probabilities that can be calculated from the sample space (HSS.MD.A.3).
Question 15 of 20 · Short Answer
A ski shop records helmet rentals for 120 groups (invented data): 0 helmets: 18 groups, 1: 30, 2: 42, 3: 24, 4: 6. Build the probability distribution of X = helmets rented by a group and find E(X).
Divide each count by 120: P = 0.15, 0.25, 0.35, 0.20, 0.05, which add to 1. E(X) = 0(0.15) + 1(0.25) + 2(0.35) + 3(0.20) + 4(0.05) = 0.25 + 0.70 + 0.60 + 0.20 = 1.75 helmets per group. Over many groups, the shop rents about 1.75 helmets per group on average.
Question 16 of 20 · Short Answer
A pizza shop's records give this distribution of X = toppings per pizza: P(0) = 0.10, P(1) = 0.45, P(2) = 0.30, P(3) = 0.15. Each topping costs the shop $0.80. Find the expected number of toppings per pizza and the expected topping cost for 300 pizzas.
E(X) = 0 + 0.45 + 0.60 + 0.45 = 1.5 toppings per pizza. The expected cost per pizza is 1.5 × $0.80 = $1.20, so for 300 pizzas the shop should expect about 300 × $1.20 = $360 in topping costs.
Question 17 of 20 · Short Answer
To estimate the distribution of X = phones per household in her town, a student asks 10 of her friends. Give two reasons her empirical distribution may be a poor estimate for the town.
Sample answer: (1) The sample is small, so the relative frequencies can change a lot by chance; each friend counts for 1/10 of the data, so one unusual household, such as one with 5 phones, moves a probability by 0.1. (2) The sample is not random: friends tend to be similar in age, neighborhood and family size, so they may not represent the town. A larger random sample of households would give a better estimate.
Question 18 of 20 · Short Answer
A store records the number of items bought by 60 customers (invented data): 1 item: 21 customers, 2 items: an unknown number, 3 items: 12, 4 items: 9. Find the missing count, the probability distribution and E(X).
The missing count is 60 - 21 - 12 - 9 = 18. Dividing by 60 gives P = 0.35, 0.30, 0.20, 0.15. E(X) = 1(0.35) + 2(0.30) + 3(0.20) + 4(0.15) = 0.35 + 0.60 + 0.60 + 0.60 = 2.15 items per customer.
Question 19 of 20 · Short Answer
A hardware store's data give an expected value of 1.6 keys copied per customer. Can a customer copy 1.6 keys? What does 1.6 mean, and how many keys should the store expect to copy for 250 customers?
No: each customer copies a whole number of keys. The value 1.6 is a long-run average: over many customers, the mean number of keys copied per customer is close to 1.6. For 250 customers, expect about 250 × 1.6 = 400 keys.
Question 20 of 20 · Short Answer
An invented table of TV sets per household reports: 0 sets: 4%, 1 set: 30%, 2 sets: 34%, 3 sets: 20%, 4 or more sets: 12%. Treat "4 or more" as 4. Find the expected number of sets per household and the number of sets you would expect in 100 randomly selected households. Is the true expected value higher or lower than your answer?
E(X) = 0(0.04) + 1(0.30) + 2(0.34) + 3(0.20) + 4(0.12) = 0.30 + 0.68 + 0.60 + 0.48 = 2.06 sets per household, so about 206 sets in 100 households. Some households in the top group have 5 or more sets, so coding them as 4 makes the result an underestimate: the true expected value is at least 2.06.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSS.MD.A.4 mean?
HSS.MD.A.4 means students build a probability distribution from data and then find its expected value. The probabilities are relative frequencies: the share of observed households, days or customers with each value of the random variable. The standard is marked (+), so it belongs to advanced courses such as Precalculus or statistics.
What is the difference between HSS.MD.A.3 and HSS.MD.A.4?
The difference is where the probabilities come from. In HSS.MD.A.3 they are theoretical: they are calculated from a sample space with known structure, such as dice or cards. In HSS.MD.A.4 they are empirical: they are estimated from observed data, such as a survey. The expected value is computed the same way in both.
How do you turn survey data into a probability distribution?
Divide the count for each value by the total number of responses. List each value of the random variable with its relative frequency, then check that every probability is between 0 and 1 and that they add to 1. If the data are given as percents, write each percent as a decimal.
Is expected value the same as the average?
Yes, for an empirical distribution the expected value equals the mean of the data it came from. Computing Σ x · P(X = x) with relative frequencies is the same as adding all the observed values and dividing by the number of observations. The expected value also predicts the long-run average for new observations, if the data represent the population.
Why can the expected value be a number the random variable never takes?
Because it is an average, not a single outcome. A household cannot own a fraction of a TV, but the mean number of sets over many households can be a decimal. Students should interpret it as "on average, per household", not "each household has".
How big should the sample be for an empirical distribution?
There is no single required size, but larger random samples give more reliable estimates. Relative frequencies from small samples change a lot by chance, and a sample that is not random can be biased at any size. Have students compare distributions from small and large samples of the same data to see the difference.
Where can students find real data for the TV sets example?
A national household survey is a good source; in the United States, the Energy Information Administration's Residential Energy Consumption Survey (RECS) reports the number of televisions in homes. Students should record the source and year, because the distribution changes over time. The data on this page are illustrative only.
What should students do with a category like "4 or more"?
Pick a value to code it with and say how that choice affects the answer. Coding "4 or more" as 4 gives a lower bound for the expected value, because some households in that group have more. Students can also try a larger value and compare, which shows how sensitive the expected value is to the top category.
Is HSS.MD.A.4 taught in Algebra 2 or in statistics?
It is usually taught in a statistics course, Precalculus or an advanced Algebra II course. As a (+) standard, it is beyond the college- and career-ready core that all students take, so some schools cover it only in an elective statistics class.
What mistakes do students make when finding expected value from data?
A common mistake is averaging the listed values without weighting them by their probabilities. Others are dividing by the number of categories instead of the number of observations, using counts as if they were probabilities, forgetting a value, and saying that every trial will produce the expected value.
07
Related Standards
6 standards
These standards connect to HSS.MD.A.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.SP.C.7Prerequisite
Develop a probability model, uniform or based on observed frequencies, and use it
