Question

At a certain store for a certain month, in a chart given below, the price per cell phone (in dollars) is shown by dots (read from right hand side vertical axis) and the number of cell phones sold (read from left hand side vertical axis).

What is the median price of the cell phones sold by the store in that month? 
 

Hint:

When dealing with charts that show frequency (like the number of items sold), the median is not simply the middle value of the unique prices. You must account for how many items were sold at each price. A common mistake is to just find the median of ($150, $200, $300, $350, $400), which would be incorrect. Always calculate the total number of items to find the correct median position.

Answer 1

Step 1: Understanding the Concept:
The median is the middle value in a dataset that has been arranged in ascending or descending order. To find the median price of all cell phones sold, we first need to determine the total number of phones sold and then find the price of the middle phone in the ordered list of all phone prices.
Step 2: Key Formula or Approach:
1. Extract the number of units sold and the price per unit for each brand from the given chart.
2. Calculate the total number of cell phones sold (N).
3. Since the number of data points will be large, we determine the position of the median. If N is an odd number, the median is the value at the \(\left( \frac{N+1}{2} \right)\)-th position.
4. Arrange the prices in ascending order and use the number of phones sold at each price to find the cumulative frequency.
5. Identify the price that corresponds to the median position.
Step 3: Detailed Explanation:
First, we read the data for each brand from the chart:

Brand P: Number sold = 75, Price = $300

Brand Q: Number sold = 75, Price = $400

Brand R: Number sold = 125, Price = $150

Brand S: Number sold = 150, Price = $350

Brand T: Number sold = 100, Price = $200

Next, we calculate the total number of cell phones sold (N):
\[ N = 75 (\text{Brand P}) + 75 (\text{Brand Q}) + 125 (\text{Brand R}) + 150 (\text{Brand S}) + 100 (\text{Brand T}) \] \[ N = 525 \] Since N = 525 (an odd number), the median is the value of the phone at the following position:
\[ \text{Median Position} = \frac{N+1}{2} = \frac{525+1}{2} = \frac{526}{2} = 263\text{-rd position} \] Now, we arrange the data in ascending order of price and find the cumulative count of phones:

Price $150 (Brand R): 125 phones sold.
(These are the 1st to 125th phones in the ordered list).
Cumulative Count = 125

Price $200 (Brand T): 100 phones sold.
(These are the 126th to 225th phones, since 125 + 100 = 225).
Cumulative Count = 225

Price $300 (Brand P): 75 phones sold.
(These are the 226th to 300th phones, since 225 + 75 = 300).
Cumulative Count = 300

Price $350 (Brand S): 150 phones sold.
Cumulative Count = 450

Price $400 (Brand Q): 75 phones sold.
Cumulative Count = 525

We are looking for the price of the 263rd phone.
- The first 125 phones cost $150.
- Phones from the 126th to the 225th position cost $200.
- Phones from the 226th to the 300th position cost $300.
Since the median position (263rd) falls between the 226th and 300th position, the median price is $300.
Step 4: Final Answer:
The median price of the cell phones sold by the store is $300.