The data below show the weights of 14 different dogs that visited a veterinarian on one day. Calculate the 49th and 91st percentiles of the data.

46	55	53	98	74	67	84	100	29	90	17	78	62	38




This means that approximately 
49
Correct% of the data lie at or below 
, when the data are ranked.



This means that approximately 
91
Correct% of the data lie at or below 
, when the data are ranked.

To calculate the 49th and 91st percentiles of the given data set, we first need to sort the data in ascending order and then use the formulas for percentiles.

### Step-by-Step Calculation:

1. **Sort the Data:**

   First, let's sort the weights in ascending order:

   \[
17, 29, 38, 46, 53, 55, 62, 67, 74, 78, 84, 90, 98, 100
   \]

2. **Determine the Position for the 49th Percentile:**

   The formula to find the position of a percentile \(P\) in a sorted list of \(n\) data points is:

   \[
\text{Position} = \frac{P}{100} \times (n + 1)
   \]

   For the 49th percentile (\(P = 49\)):

   \[
\text{Position} = \frac{49}{100} \times (14 + 1) = 0.49 \times 15 = 7.35
   \]

   The position is 7.35, which is between the 7th and 8th values.

3. **Calculate the 49th Percentile:**

   The 49th percentile will be between the 7th and 8th values in the sorted list. We use linear interpolation:

   - 7th value = 62
   - 8th value = 67

   The fractional part is 0.35 (from 7.35), so:

   \[
\text{49th Percentile} = 62 + 0.35 \times (67 - 62) = 62 + 0.35 \times 5 = 62 + 1.75 = 63.75
   \]

   Therefore, the 49th percentile is **63.75**.

4. **Determine the Position for the 91st Percentile:**

   For the 91st percentile (\(P = 91\)):

   \[
\text{Position} = \frac{91}{100} \times (14 + 1) = 0.91 \times 15 = 13.65
   \]

   The position is 13.65, which is between the 13th and 14th values.

5. **Calculate the 91st Percentile:**

   The 91st percentile will be between the 13th and 14th values in the sorted list. Again, we use linear interpolation:

   - 13th value = 98
   - 14th value = 100

   The fractional part is 0.65 (from 13.65), so:

   \[
\text{91st Percentile} = 98 + 0.65 \times (100 - 98) = 98 + 0.65 \times 2 = 98 + 1.3 = 99.3
   \]

   Therefore, the 91st percentile is **99.3**.

### Summary of Results:

- **49th Percentile:** 63.75  
- **91st Percentile:** 99.3  

These percentiles mean:

- **49%** of the data lie at or below **63.75**.
- **91%** of the data lie at or below **99.3**.

Would you like more details or have any questions?

### Related Questions:

1. How do you calculate the median for this set of data?
2. What is the 25th percentile of this data set?
3. How would the percentiles change if we added more data points?
4. Can you explain the difference between quartiles and percentiles?
5. How is the interquartile range (IQR) calculated for this data?

**Tip:** Percentiles are useful for understanding the distribution of data, particularly in identifying the spread and skewness of data points.

Learn how to calculate the 49th and 91st percentiles of a dataset containing dog weights. This detailed guide includes sorting data, using percentile formulas, and applying linear interpolation. Suitable for students studying descriptive statistics and percentiles.

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