If the algebraic sum of the deviations of twelve observations measured from 40 is 60, then find their AM

Given that the algebraic sum of the deviations of twelve observations measured from 40 is 60, we need to find the arithmetic mean (AM) of the observations.

### Step 1: Understanding the concept
The algebraic sum of deviations of a set of observations from their arithmetic mean (AM) is always zero. If \( x_1, x_2, \ldots, x_n \) are the observations, and \( \overline{x} \) is their arithmetic mean, then:

\[
\sum_{i=1}^n (x_i - \overline{x}) = 0
\]

However, in this problem, we are given the algebraic sum of the deviations from 40, not from the actual mean.

### Step 2: Expressing the information given
Let's denote the twelve observations by \( x_1, x_2, \ldots, x_{12} \).

The algebraic sum of deviations from 40 is given by:

\[
\sum_{i=1}^{12} (x_i - 40) = 60
\]

### Step 3: Simplifying the equation
Expanding the sum:

\[
\sum_{i=1}^{12} x_i - 12 \times 40 = 60
\]

This simplifies to:

\[
\sum_{i=1}^{12} x_i - 480 = 60
\]

Adding 480 to both sides:

\[
\sum_{i=1}^{12} x_i = 540
\]

### Step 4: Calculating the Arithmetic Mean (AM)
The arithmetic mean (AM) is given by:

\[
\overline{x} = \frac{\sum_{i=1}^{12} x_i}{12} = \frac{540}{12} = 45
\]

### Final Answer:
The arithmetic mean (AM) of the twelve observations is **45**.

Would you like further details or have any other questions?

### Follow-up Questions:

1. What is the definition of the arithmetic mean, and how is it different from the median?
2. If the deviations were measured from a different value, how would that affect the mean?
3. How would you find the median of the same set of observations?
4. Can you determine the standard deviation with the given information?
5. How does the sum of deviations being zero relate to the arithmetic mean?

**Tip:** Always remember that the sum of deviations from the arithmetic mean is zero, a useful property when solving statistical problems.

Learn how to find the arithmetic mean (AM) of twelve observations given their deviations from a reference value. This problem involves understanding deviations and calculating the mean, suitable for high school level students.

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