Let's solve the problem step-by-step based on the data provided:

The data given:

2, 3, 1, 2, 6, 4, 2, 1, 5, 3, 3, 2, 1, 3, 1, 2, 4, 2, 1, 2, 8, 3, 3, 2, 1, 3, 3, 2, 1, 3

1. Mean

The mean (average) is calculated as:

$\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}$

• Sum of the data points: $2 + 3 + 1 + 2 + 6 + 4 + 2 + 1 + 5 + 3 + 3 + 2 + 1 + 3 + 1 + 2 + 4 + 2 + 1 + 2 + 8 + 3 + 3 + 2 + 1 + 3 + 3 + 2 + 1 + 3 = 74$
• Number of data points: $30$

$\text{Mean} = \frac{74}{30} \approx 2.47$

2. Median

The median is the middle value when the data is sorted in ascending order.

• Sorted data:
  1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5, 6, 8

• Since there are 30 data points (even number), the median is the average of the 15th and 16th values in the sorted list.

15th value = 2
16th value = 2

$\text{Median} = \frac{2 + 2}{2} = 2$

3. Mode

The mode is the value that appears most frequently.

• Frequency table:
  1 appears 6 times
  2 appears 10 times
  3 appears 8 times
  4 appears 2 times
  5 appears 1 time
  6 appears 1 time
  8 appears 1 time

The mode is 2 since it appears the most frequently (10 times).

4. Range

The range is the difference between the largest and smallest values.

$\text{Range} = \text{Maximum} - \text{Minimum} = 8 - 1 = 7$

5. Lower and Upper Quartiles (Q1 and Q3)

• Q1 (lower quartile) is the median of the first half of the sorted data (lower 15 values).
• Q3 (upper quartile) is the median of the second half of the sorted data (upper 15 values).

Lower half (first 15 values):

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2
Q1 = median of these values = 2

Upper half (last 15 values):

2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5, 6, 8
Q3 = median of these values = 3

6. Interquartile Range (IQR)

The IQR is the difference between Q3 and Q1:

$\text{IQR} = Q3 - Q1 = 3 - 2 = 1$

Final Results

1. Mean: $2.47$
2. Median: $2$
3. Mode: $2$
4. Range: $7$
5. Q1: $2$, Q3: $3$
6. Interquartile Range (IQR): $1$

Do you want me to explain any step in detail? Let me know!
Here are some related questions for practice:

1. How is the mean affected by an outlier in the data?
2. What is the difference between mean and median?
3. Can a data set have more than one mode?
4. How do you calculate the IQR for an odd number of data points?
5. What does the range tell us about a data set?

Tip: The IQR (Interquartile Range) is a robust measure of spread because it is not affected by extreme values (outliers).