To evaluate whether the highlighted part is correct, let's break down the calculation of the 7th decile (D7) step by step.

Given Data:

• m = 7 (indicating the decile number to calculate, i.e., D7)
• K = 10 (K refers to the total number of deciles, meaning we're dividing the data into 10 equal parts)
• N = 11 (the total number of data points)

Formula for the Decile (Dk):

The formula for the $k$-th decile is: $D_k = \left( \frac{k}{K} \right) \times N$ Where:

• $k$ is the decile number (in this case, 7)
• $K = 10$, since deciles divide the data into 10 parts
• $N = 11$, the total number of data points

Plugging in the values for the 7th decile: $D_7 = \left( \frac{7}{10} \right) \times 11 = 7.7$

Interpretation of 7.7:

The number 7.7 represents the position of the 7th decile within the ordered data set. Since deciles split the data into 10 equal parts, the 7th decile will lie between the 7th and 8th data points when the data is sorted in ascending order.

Ordered Data:

The sorted data set is: $5, 7, 7, 8, 8, 9, 9, 10, 10, 12, 12, 13$

• The 7th data point is 9
• The 8th data point is 10

Since 7.7 lies between the 7th and 8th positions, we can interpret the 7th decile as a value between 9 and 10. Depending on how interpolation is handled, the decile could be taken as either of these numbers or an average.

Conclusion:

The highlighted calculation of $\left( \frac{7}{10} \right) \times 11 = 7.7$ is correct.

Would you like a more detailed explanation of the interpolation or any other part?

Expanding Questions:

1. How is the interpretation of deciles different from quartiles or percentiles?
2. What would the result be if the total number of data points $N$ were different, say 20?
3. Could we calculate the 5th decile (the median) using the same formula? How?
4. How can we improve the accuracy of interpreting deciles for small datasets like this?
5. How would you calculate other deciles (e.g., the 3rd or 9th decile) for this dataset?

Tip:

For more precision in interpreting deciles, consider interpolation techniques when the index is a decimal value.