Question
PRICE OF A VIDEO GAME AT VARIOUS STORES
| Price of Video Game | Number of Stores |
|---|---|
| $10.99 | 5 |
| $11.99 | 2 |
| $12.99 | 1 |
| $13.99 | 4 |
The table above shows the price of a certain video game at 12 different stores. What is the median price of this video game?
- A$10.99
- B$11.49
- C$11.99
- D$12.32
Show the answer
Correct answer: C
There are 5 stores where the video game is priced at $10.99 and a total of 5 stores where it is priced at either $12.99 or $13.99, so the median price must be $11.99.
Question analysis
A frequency table gives the price of one video game at 12 stores, from $10.99 to $13.99, with the number of stores charging each price. Students find the median price. This statistics question tests the median, the middle of an ordered data set, when the data arrive as a table of counts rather than a list.
- Grade level
- Grade 6 content, by NYS standard
- Compared with grade 7
- At grade 7 The median is a grade 6 measure of center that grade 7 students use regularly. What lifts this slightly is the table of counts: students must work out which rows hold the middle prices of the 12.
- Estimated difficulty
- Straightforward: two steps or one careful reading.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Solution
- 2 steps
- Reading level
- Question and choices: about grade 4, 37 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
Once the middle positions are found there is nothing to compute: the first 5 stores charge $10.99, and the next two, the 6th and 7th, both charge $11.99. The wrong choices each reward a different misuse of a frequency table, taking the most common price, averaging the 5th and 6th values, or working out the mean, which costs time and still misses. Counting stores down the table, 5 and then 7, shows at once that both middle positions fall in the second row.
Reading a figure, table or graph moderate Understanding the question
Each row stands for several stores, 5 at $10.99, 2 at $11.99, 1 at $12.99 and 4 at $13.99, so the ordered list of 12 prices exists only in the student's head. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Something you must already know minor Working it out
The median of an even number of values is the mean of the two middle ones, here the 6th and 7th; the question assumes this, though both turn out to cost $11.99. (Anderson & Krathwohl, 2001)
Strategy: With 12 prices, the median is the mean of the 6th and 7th once they are in order; add up the store counts from the cheapest price until you reach 7.
Traps
- Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
- Counting the ends: Inclusive or exclusive ends, fence posts, consecutive numbers: an answer that is one too many or too few, or an endpoint drawn open where it belongs closed (or the reverse).
The wrong answers, classified
A Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Picks $10.99 because more stores charge it than any other price, giving the mode in place of the middle price of the 12.
B Counting the ends (Brown & Burton, 1978)
Averages two middle prices but takes the 5th and 6th, $10.99 and $11.99, when the middle pair of 12 values is the 6th and 7th.
D Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Works out the mean, $147.88 for the 12 stores divided by 12, about $12.32, a measure of center the question does not ask for.
Worked solution, step by step
Formulas used
- Middle places of an even count
- n is how many values there are, when n is an even number. Line the values up in order. The two values in these places have the same number of values before them as after them. For 10 values, they are the 5th and 6th.
- Median of an even number of values
- The median is the middle of a list of values put in order from least to greatest. With an even number of values, two of them share the middle. The median is their mean: add them and divide by 2.
Sort out the question
The table shows one video game’s price at 12 stores. Each row gives a price and how many stores charge it. For example, 4 stores charge $13.99, so $13.99 is in the list 4 times. You need the median price. That is the middle price when all 12 prices are lined up from least to greatest.
Find the two middle places
First check the number of stores: . That is an even number, so no one price sits in the middle. Two prices share it.
The middle prices are the 6th and 7th. Five prices come before the 6th, and five come after the 7th.
Count down the table
The table already lists the prices from least to greatest. Keep a running count of stores as you go down:
- $10.99: 5 stores, places 1 to 5
- $11.99: 2 stores, places 6 and 7
- $12.99: 1 store, place 8
- $13.99: 4 stores, places 9 to 12
Both middle places, the 6th and the 7th, hold $11.99.
Find the median
Use the median formula with the 6th and 7th prices.
The median price is $11.99.
Watch out: $10.99 is the price that shows up most often, called the mode. That is choice A. Averaging the 5th and 6th prices gives $11.49, choice B. Adding all 12 prices and dividing by 12 gives the mean, about $12.32, choice D.
So the answer is C, $11.99.
Check your answer
Look at each side of $11.99. Cheaper: the 5 stores at $10.99. More expensive: 1 store at $12.99 and 4 at $13.99, which is also 5 stores. Five below and five above, with the two $11.99 stores between them. So $11.99 is in the middle.
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