Let's solve the problem step by step for the given data set:

Data set: 5, 6, 6, 9, 12, 12, 15, 12, 57, 45, 35, 28, 12

1. Arrange the data in ascending order: 5, 6, 6, 9, 12, 12, 12, 12, 15, 28, 35, 45, 57

Calculations:

1. Mean: Formula: $\text{Mean} = \frac{\text{Sum of all data points}}{\text{Total number of data points}}$

   $\text{Mean} = \frac{5 + 6 + 6 + 9 + 12 + 12 + 12 + 12 + 15 + 28 + 35 + 45 + 57}{13} = \frac{234}{13} \approx 18$

2. Median: The median is the middle value in the ordered data set. For $n = 13$ (odd), the median is the 7th value.

   Median = 12

3. Mode: The mode is the most frequently occurring value. In this data set, 12 occurs 4 times.

   Mode = 12

4. 1st Quartile (Q1): Formula: $Q_1 = \text{Median of the lower half of data}$

   Lower half: 5, 6, 6, 9, 12, 12 Median of 5, 6, 6, 9, 12, 12 = $\frac{6 + 9}{2} = 7.5$

   $Q_1 = 7.5$

5. 3rd Quartile (Q3): Formula: $Q_3 = \text{Median of the upper half of data}$

   Upper half: 12, 15, 28, 35, 45, 57 Median of 12, 15, 28, 35, 45, 57 = $\frac{28 + 35}{2} = 31.5$

   $Q_3 = 31.5$

6. 4th Decile (D4): Formula: $D_k = \text{Value at } \frac{k(n+1)}{10}$th position

   $D_4 = \text{Value at } \frac{4(13+1)}{10} = \frac{4 \times 14}{10} = 5.6$th position.

   The 5.6th position is between the 5th and 6th data points (12 and 12). $D_4 = 12$

7. 40th Percentile (P40): Formula: $P_k = \text{Value at } \frac{k(n+1)}{100}$th position

   $P_{40} = \text{Value at } \frac{40(13+1)}{100} = \frac{40 \times 14}{100} = 5.6$th position.

   Similar to $D_4$, $P_{40} = 12$

Final Results:

1. Mean: 18
2. Median: 12
3. Mode: 12
4. 1st Quartile (Q1): 7.5
5. 3rd Quartile (Q3): 31.5
6. 4th Decile (D4): 12
7. 40th Percentile (P40): 12

Would you like a detailed explanation of any calculation or further visual representation?

5 Related Questions:

1. How are quartiles and percentiles different from each other?
2. Can the mean and median of a data set be the same? If so, when?
3. How do outliers (like 57 here) affect the mean compared to the median?
4. How would you calculate the interquartile range (IQR) for this data set?
5. How is the mode useful in real-world data analysis?

Tip:

When a data set has extreme outliers, the median is a more reliable measure of central tendency than the mean.