Question

Define Average value and give its mathematical expression.

Answer 1

The average value, also known as the arithmetic mean, of a set of values is a measure of central tendency that provides a single value representing the center of the distribution of those values.

### Definition

For a given set of \( n \) values \( x_1, x_2, x_3, \ldots, x_n \), the average value is the sum of these values divided by the number of values.

### Mathematical Expression

The mathematical expression for the average value \( \bar{x} \) is:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]

where:
- \( \bar{x} \) is the average value,
- \( n \) is the number of values in the set,
- \( x_i \) represents each individual value in the set,
- \( \sum_{i=1}^{n} x_i \) is the sum of all values in the set.

### Example

Consider a set of values: 4, 8, 15, 16, 23, 42.

1. Calculate the sum of the values:

\[ 4 + 8 + 15 + 16 + 23 + 42 = 108 \]

2. Count the number of values:

\[ n = 6 \]

3. Compute the average value:

\[ \bar{x} = \frac{108}{6} = 18 \]

So, the average value of this set is 18.

Answer 2

The average value of a function over a given interval represents the mean value that the function takes over that interval. For a continuous function \( f(x) \) defined on the interval \([a, b]\), the average value is given by:

\[ \text{Average Value} = \frac{1}{b - a} \int_a^b f(x) \, dx \]

This formula calculates the mean value of \( f(x) \) over \([a, b]\) by integrating \( f(x) \) across the interval and then dividing by the length of the interval \((b - a)\).