Let's compute the mean, median, mode, and range for the following data set of observations:

Data Set: $7, 12, 4, 8, 3, 10, 11, 5, 13, 1, 12, 3, 5, 1, 17, 4, 8, 8, 7, 10, 8, 1, 7, 17, 4, 7, 1, 7, 3, 7, 3, 13, 3, 4, 7, 8, 10, 2, 5, 11, 5, 4, 3, 5, 8$

Step 1: Mean

The mean is the average of the data. To find the mean, sum all the observations and divide by the number of observations.

$\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}$

Let's calculate that:

$\text{Sum of observations} = 7 + 12 + 4 + 8 + 3 + 10 + 11 + 5 + 13 + 1 + 12 + 3 + 5 + 1 + 17 + 4 + 8 + 8 + 7 + 10 + 8 + 1 + 7 + 17 + 4 + 7 + 1 + 7 + 3 + 7 + 3 + 13 + 3 + 4 + 7 + 8 + 10 + 2 + 5 + 11 + 5 + 4 + 3 + 5 + 8$

Now, sum the values and divide by the total count of numbers (45 values).

$\text{Sum} = 289$ $\text{Mean} = \frac{289}{45} \approx 6.42$

Step 2: Median

The median is the middle value when the data is ordered from least to greatest. If the number of observations is odd, the median is the middle number. If even, it's the average of the two middle numbers.

First, let's arrange the data in ascending order:

$1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 11, 11, 12, 12, 13, 13, 17, 17$

The data set has 45 observations (an odd number), so the median is the 23rd value.

$\text{Median} = 5$

Step 3: Mode

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

Looking at the ordered data, the number that appears most frequently is $7$ (which appears 8 times).

$\text{Mode} = 7$

Step 4: Range

The range is the difference between the maximum and minimum values in the data set.

$\text{Range} = \text{Maximum} - \text{Minimum} = 17 - 1 = 16$

Summary:

• Mean: $6.42$
• Median: $5$
• Mode: $7$
• Range: $16$

Would you like more details on any of these calculations or have any further questions?

Related Questions:

1. How would the mean change if we add more data points?
2. What is the difference between the median and mode for skewed data?
3. How does the presence of outliers affect the range?
4. Can a data set have more than one mode?
5. How can we calculate the mean and median for grouped data?

Tip:

For a data set with extreme values, the median can be a better representation of the "typical" value than the mean!