We will solve this problem step by step.

Given the tensile strength dataWe will solve this problem step by step.

Given the tensile strength

18, 13, 2, 20, 8, 10, 5, 10, 6, 9, 10, 20, 2, 15, 16, 16, 13, 10, 17, 10, 3, 2, 15, 8, 5

---

1) Calculate the Mean, Median, and Mode:

Step 1: Mean

The formula for mean is: $\text{Mean} = \frac{\sum x_i}{n}$ Where $\sum x_i$ is the sum of all observations and $n$ is the number of observations.

Step 2: Median

To find the median, we need to sort the data in ascending order and find the middle value(s).

Step 3: Mode

The mode is the number that appears most frequently in the data set.

---

2) Show that Sum of Deviations from Mean is Zero:

For any data set, the sum of the deviations from the mean is always zero, i.e.: $\sum (x_i - \mu) = 0$ Where $x_i$ are the individual data points and $\mu$ is the mean.

---

3) Calculate Quartiles, Seventh Decile (D7), P30, and P75:

• Quartiles (Q1, Q2, Q3): These divide the data into four equal parts.
• Deciles (D7): These divide the data into 10 equal parts.
• Percentiles (P30, P75): These divide the data into 100 equal parts.

Let me calculate each step to provide the exact values.Here are the calculations:

1) Mean, Median, and Mode:

• Mean: $10.52$ (rounded to two decimal places)
• Median: $10.0$
• Mode: $10$ (appears most frequently)

2) Sum of Deviations from Mean:

The sum of deviations from the mean is approximately $0$ (due to rounding, it is $1.07 \times 10^{-14}$, which is essentially zero).

3) Quartiles, Seventh Decile (D7), and Percentiles (P30, P75):

• Q1 (25th percentile): $6.0$
• Q2 (50th percentile or median): $10.0$
• Q3 (75th percentile): $15.0$
• Seventh Decile (D7 or 70th percentile): $14.6$
• P30 (30th percentile): $8.0$
• P75 (75th percentile): $15.0$

These values give a good summary of the data distribution.

---

Let me know if you'd like more details on these steps or have questions! Here are 5 related questions to consider:

1. How is the mode useful in understanding data distribution?
2. Why is the sum of deviations from the mean always zero?
3. How do quartiles help in summarizing data spread?
4. What is the difference between percentiles and deciles?
5. Why is the median a more robust measure of central tendency than the mean?

Tip: Quartiles, deciles, and percentiles are all useful for understanding the spread and central values in a data set, but they focus on different levels of granularity.