Analyzing Dice Observations and Calculating Means

The problem provides a set of 12 observations, which are the results of throwing 12 dice simultaneously. We are asked to find the mean of the lowest 8 observations (denoted by 'm') and the mean of the highest 4 observations (denoted by 'M'), and then calculate the value of $2m + M$.

Sorting the Observations

To identify the lowest 8 and highest 4 observations, we first need to sort the given data in ascending order.

The given observations are: 4, 1, 4, 3, 6, 2, 1, 3, 4, 5, 1, 6.

Let's list the observations:

• 4
• 1
• 4
• 3
• 6
• 2
• 1
• 3
• 4
• 5
• 1
• 6

Sorting these observations in ascending order, we get:

• 1
• 1
• 1
• 2
• 3
• 3
• 4
• 4
• 4
• 5
• 6
• 6

Identifying Lowest 8 Observations and Calculating Mean 'm'

The lowest 8 observations from the sorted list are the first 8 values:

1, 1, 1, 2, 3, 3, 4, 4

The mean 'm' of these 8 observations is the sum of these observations divided by the number of observations (8).

Sum of lowest 8 observations $= 1 + 1 + 1 + 2 + 3 + 3 + 4 + 4 = 19$

Number of lowest observations $= 8$

Mean $m = \frac{\text{Sum of lowest 8 observations}}{\text{Number of lowest observations}}$

Using the formula for mean:

$\qquad m = \frac{19}{8}$

Identifying Highest 4 Observations and Calculating Mean 'M'

The highest 4 observations from the sorted list are the last 4 values:

4, 5, 6, 6

The mean 'M' of these 4 observations is the sum of these observations divided by the number of observations (4).

Sum of highest 4 observations $= 4 + 5 + 6 + 6 = 21$

Number of highest observations $= 4$

Mean $M = \frac{\text{Sum of highest 4 observations}}{\text{Number of highest observations}}$

Using the formula for mean:

$\qquad M = \frac{21}{4}$

Calculating the Value of (2m + M)

Now, we need to calculate the value of the expression $2m + M$. We have the values for m and M.

$m = \frac{19}{8}$

$M = \frac{21}{4}$

Let's substitute these values into the expression:

$2m + M = 2 \times \left(\frac{19}{8}\right) + \frac{21}{4}$

First, calculate $2m$:

$2 \times \frac{19}{8} = \frac{2 \times 19}{8} = \frac{38}{8} = \frac{19}{4}$

Now, substitute this back into the expression $2m + M$:

$2m + M = \frac{19}{4} + \frac{21}{4}$

Since the fractions have a common denominator (4), we can simply add the numerators:

$2m + M = \frac{19 + 21}{4} = \frac{40}{4}$

Finally, simplify the fraction:

$2m + M = 10$

So, the value of $(2m + M)$ is 10.

Revision Table: Key Concepts

Additional Information on Data Analysis

Analyzing data often begins with organizing it. Sorting, as done in this problem, helps in quickly identifying minimum, maximum, and ranges of values. Measures of central tendency like the mean, median, and mode provide a single value that represents the center of the data distribution. The mean is sensitive to extreme values, while the median (the middle value of sorted data) is not.

In problems involving subsets of data, like the lowest or highest values, sorting is a crucial first step before applying statistical calculations. Understanding how to perform basic operations like sorting and calculating means is fundamental in statistics and data analysis.