Let's solve the problem based on the data in the uploaded image.

Step 1: Gather the data

The number of yo-yos per penguin is represented by the following frequencies:

• 3 yo-yos: 1 penguin
• 5 yo-yos: 4 penguins
• 6 yo-yos: 2 penguins
• 7 yo-yos: 3 penguins
• 8 yo-yos: 2 penguins
• 9 yo-yos: 2 penguins

Step 2: Organize the data

List all the values with their frequencies: 3, 5, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9.

Step 3: Calculate the required values

(a) Range

$\text{Range} = \text{Maximum} - \text{Minimum} = 9 - 3 = 6.$ Range = 6.

(b) Mean

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of values}}.$

• Sum of all values: $3 + (5 \times 4) + (6 \times 2) + (7 \times 3) + (8 \times 2) + (9 \times 2) = 3 + 20 + 12 + 21 + 16 + 18 = 90.$

• Total number of values: $1 + 4 + 2 + 3 + 2 + 2 = 14.$

$\text{Mean} = \frac{90}{14} \approx 6.4.$ Mean = 6.4.

(c) Median

The median is the middle value of the ordered dataset. Since there are 14 values (an even number), the median is the average of the 7th and 8th values.

• Ordered dataset:
  3, 5, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9.
  The 7th and 8th values are 6 and 7.

$\text{Median} = \frac{6 + 7}{2} = 6.5.$ Median = 6.5.

(d) Mode

The mode is the most frequently occurring value in the dataset.

• Frequency count:
  • 3 appears 1 time.
  • 5 appears 4 times.
  • 6 appears 2 times.
  • 7 appears 3 times.
  • 8 appears 2 times.
  • 9 appears 2 times.

The mode is 5 (it appears 4 times, the most frequent).

Mode = 5.

Final Answers:

• Range: 6.
• Mean: 6.4.
• Median: 6.5.
• Mode: 5.

Would you like a step-by-step explanation for any part?

5 Related Questions:

1. How do you compute the range for a different dataset?
2. What happens to the mean if an outlier is added to this data?
3. Can a dataset have more than one mode? Provide an example.
4. What is the difference between mean and median in symmetric vs. skewed datasets?
5. Why is the range not always the best measure of variability?

Tip:

When finding the median, always double-check the total number of values to ensure proper placement of the middle term(s).