13    14    14    16    16    16    16    18    18   19


a. What is the mean?  
 centimeters


b. What is the MAD?  
 centimeters

Let's break down the problem step by step.

### Part a: Calculating the Mean

The mean (average) is calculated by summing all the numbers and then dividing by the total number of values.

Given dataLet's break down the problem step by step.

### Part a: Calculating the Mean

The mean (average) is calculated by summing all the numbers and then dividing by the total number of values.

Given  13, 14, 14, 16, 16, 16, 16, 18, 18, 19

1. **Sum of the values**:
   \[
   13 + 14 + 14 + 16 + 16 + 16 + 16 + 18 + 18 + 19 = 160
   \]

2. **Total number of values**:
   \[
   10
   \]

3. **Mean**:
   \[
   \text{Mean} = \frac{160}{10} = 16
   \]
So, the mean is **16 centimeters**.

### Part b: Calculating the Mean Absolute Deviation (MAD)

The Mean Absolute Deviation (MAD) measures the average distance between each data point and the mean.

1. **Find the absolute deviations** from the mean (16):
   - |13 - 16| = 3
   - |14 - 16| = 2
   - |14 - 16| = 2
   - |16 - 16| = 0
   - |16 - 16| = 0
   - |16 - 16| = 0
   - |16 - 16| = 0
   - |18 - 16| = 2
   - |18 - 16| = 2
   - |19 - 16| = 3

2. **Sum of absolute deviations**:
   \[
   3 + 2 + 2 + 0 + 0 + 0 + 0 + 2 + 2 + 3 = 14
   \]

3. **MAD**:
   \[
   \text{MAD} = \frac{14}{10} = 1.4
   \]
So, the MAD is **1.4 centimeters**.

---

Would you like more details or have any questions?

Here are five related questions:
1. How would the mean change if an additional value of 20 was added to the dataset?
2. Can you calculate the variance and standard deviation for the same dataset?
3. What would the median of this dataset be?
4. How does the range of this dataset compare to the MAD?
5. What impact does a higher MAD have on the spread of the data?

**Tip:** The Mean Absolute Deviation is useful for understanding how spread out your data points are around the mean, giving insight into the consistency of the data.

Learn how to calculate the mean and mean absolute deviation (MAD) for a dataset of numbers. This problem involves finding the average value and measuring the average distance of data points from the mean, providing insights into data consistency. Suitable for high school students studying statistics.

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