Question

Look at the strings data. {0, 10, 40, 40, 40, 50, 50, 50, 60, 60} Find the median and

mode. Is the mean greater than or less than the median? Explain.

Answer 1

To find the median and mode of the dataset {0, 10, 40, 40, 40, 50, 50, 50, 60, 60}, let's first organize the data and calculate:

### Median:
1. **Sort the data** (already sorted): {0, 10, 40, 40, 40, 50, 50, 50, 60, 60}
2. **Count the number of elements**: There are 10 numbers in the dataset, which is even.
3. **Find the middle elements**: The middle elements are the 5th and 6th values in the sorted list:
- 5th value: 40
- 6th value: 50
4. **Calculate the median**:
\[
\text{Median} = \frac{40 + 50}{2} = \frac{90}{2} = 45
\]

### Mode:
1. **Identify the most frequent numbers**:
- 40 appears 3 times.
- 50 appears 3 times.
- 0, 10, and 60 appear 1 time each.
2. Since both 40 and 50 occur most frequently (3 times each), the mode is:
- **Mode**: 40 and 50 (the dataset is multimodal).

### Mean:
1. **Calculate the sum of all elements**:
\[
0 + 10 + 40 + 40 + 40 + 50 + 50 + 50 + 60 + 60 = 400
\]
2. **Count the number of elements**: 10
3. **Calculate the mean**:
\[
\text{Mean} = \frac{\text{Sum}}{\text{Number of elements}} = \frac{400}{10} = 40
\]

### Comparison of Mean and Median:
- **Mean**: 40
- **Median**: 45

Since the mean (40) is less than the median (45), **the mean is less than the median**.

### Explanation:
In this dataset, the presence of higher values (60) does not significantly skew the mean because they are not outliers. The median represents the middle value, which can be more robust to the effects of extreme values compared to the mean. However, since this dataset contains repeat values, the mean ends up being dragged down somewhat by the lower values (such as 0 and 10).