What is the five number summary for the data set : 129, 156, 130, 158, 134, 158, 139, 159, 141, 160, 143, 165, 146, 168, 149, 168, 151, 172, 152, 178

The five-number summary consists of the following elements: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum.

To calculate the five-number summary for the given data set:

**Data set:** 129, 156, 130, 158, 134, 158, 139, 159, 141, 160, 143, 165, 146, 168, 149, 168, 151, 172, 152, 178

### Steps:
1. **Sort the dataThe five-number summary consists of the following elements: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum.

To calculate the five-number summary for the given data set:

**Data set:** 129, 156, 130, 158, 134, 158, 139, 159, 141, 160, 143, 165, 146, 168, 149, 168, 151, 172, 152, 178

### Steps:
1. **Sort the **
   129, 130, 134, 139, 141, 143, 146, 149, 151, 152, 156, 158, 158, 159, 160, 165, 168, 168, 172, 178

2. **Minimum (the smallest number):**
   - **129**

3. **First Quartile (Q1, the median of the lower half):**
   - Lower half: 129, 130, 134, 139, 141, 143, 146, 149, 151, 152
   - Median of the lower half (Q1) is the average of 141 and 143:
     \[
     \frac{141 + 143}{2} = 142
     \]

4. **Median (middle value of the entire data set):**
   - Data set: 129, 130, 134, 139, 141, 143, 146, 149, 151, 152, 156, 158, 158, 159, 160, 165, 168, 168, 172, 178
   - The median is the average of 152 and 156:
     \[
     \frac{152 + 156}{2} = 154
     \]

5. **Third Quartile (Q3, the median of the upper half):**
   - Upper half: 156, 158, 158, 159, 160, 165, 168, 168, 172, 178
   - Median of the upper half (Q3) is the average of 165 and 168:
     \[
     \frac{165 + 168}{2} = 166.5
     \]

6. **Maximum (the largest number):**
   - **178**

### Five-number summary:
- **Minimum:** 129
- **Q1:** 142
- **Median:** 154
- **Q3:** 166.5
- **Maximum:** 178

Would you like a more detailed explanation of any of these steps?

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Here are five related questions to consider:
1. How would the five-number summary change if an additional value was added to the data set?
2. What is the interquartile range (IQR) for this data set?
3. How can we use the five-number summary to detect outliers in the data set?
4. Can you calculate the mean of this data set and compare it to the median?
5. What are the implications of a skewed distribution in relation to the five-number summary?

**Tip:** The interquartile range (IQR) is a good measure of variability in the data, and it is used to identify outliers.

Calculate the five-number summary for the data set: 129, 156, 130, 158, 134, 158, 139, 159, 141, 160, 143, 165, 146, 168, 149, 168, 151, 172, 152, 178. The five-number summary includes the minimum, Q1, median, Q3, and maximum. Learn step-by-step how to find these values.

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