To solve the problem, we need to find the median, mode, and average (arithmetic mean) of the scores of 21 people who played a game. Let's go through the steps one by one.
### Step 1: List the Scores
We have the following scores based on the number of people who scored them:
- 1 person scored 50 points
- 2 people scored 60 points
- 3 people scored 70 points
- 4 people scored 80 points
- 5 people scored 90 points
- 6 people scored 100 points
We can list all the scores in ascending order:
- 50 (1 time)
- 60 (2 times)
- 70 (3 times)
- 80 (4 times)
- 90 (5 times)
- 100 (6 times)
Thus, the complete list of scores is:
50, 60, 60, 70, 70, 70, 80, 80, 80, 80, 90, 90, 90, 90, 90, 100, 100, 100, 100, 100, 100
### Step 2: Calculate the Average (Arithmetic Mean)
To find the average, we need to sum all the scores and divide by the total number of scores.
**Sum of Scores Calculation:**
- \( 1 \times 50 = 50 \)
- \( 2 \times 60 = 120 \)
- \( 3 \times 70 = 210 \)
- \( 4 \times 80 = 320 \)
- \( 5 \times 90 = 450 \)
- \( 6 \times 100 = 600 \)
Now, add these values together:
\[
50 + 120 + 210 + 320 + 450 + 600 = 1750
\]
**Average Calculation:**
\[
\text{Average} = \frac{\text{Total Sum of Scores}}{\text{Total Number of Scores}} = \frac{1750}{21} \approx 83.33
\]
### Step 3: Calculate the Mode
The mode is the score that occurs most frequently. From our list of scores:
- 100 occurs 6 times (the highest frequency).
Thus, the mode is:
\[
\text{Mode} = 100
\]
### Step 4: Calculate the Median
To find the median, we need to determine the middle score in our ordered list. Since there are 21 scores (an odd number), the median is the score at the position:
\[
\text{Median Position} = \frac{n + 1}{2} = \frac{21 + 1}{2} = 11
\]
Now, we count to the 11th score in our ordered list:
1. 50
2. 60
3. 60
4. 70
5. 70
6. 70
7. 80
8. 80
9. 80
10. 80
11. 90
Thus, the median is:
\[
\text{Median} = 90
\]
### Step 5: Order the Values
Now we have:
- Average = 83.33
- Median = 90
- Mode = 100
Ordering these values from smallest to largest:
1. Average: 83.33
2. Median: 90
3. Mode: 100
### Final Answer
The correct order of the median, mode, and average is:
\[
\text{Average} < \text{Median} < \text{Mode}
\]
