Understanding Averages: Measures of Central Tendency
Hi everyone! I’m Bindeshwar from Post Network Academy. In this post, we’ll dive into averages, also known as measures of central tendency. These concepts help us understand the central part of data distribution, giving us a clearer picture of the data we’re working with.
What Are Measures of Central Tendency?
Measures of central tendency include several key averages:
1. Arithmetic Mean
2. Median
3. Mode
4. Geometric Mean
5. Harmonic Mean
Today, we’ll focus specifically on the **Arithmetic Mean**. In future videos, I’ll cover the median, mode, geometric mean, and harmonic mean in detail.
Understanding Arithmetic Mean
The arithmetic mean is a common way to calculate the average of a set of observations. There are two methods to compute it:
Direct Method
If we have \( n \) observations, denoted as \( X_1, X_2, X_3, \ldots, X_n \), the formula for the arithmetic mean \( \bar{X}_{ar} \) is:
\[
\bar{X}_{ar} = \frac{X_1 + X_2 + X_3 + \ldots + X_n}{n}
\]
Example:
Let’s say our observations are: 54, 56, 70, 45, and 50.
To calculate the mean:
\[
\bar{X}_{ar} = \frac{54 + 56 + 70 + 45 + 50}{5} = \frac{275}{5} = 55
\]
2. Shortcut Method
The shortcut method is often easier. Here’s how it works:
1. Choose an arbitrary point \( C \) (assumed value).
2. Use the formula:
\[
\bar{X}_{ar} = C + \frac{\sum_{i=1}^{n} d_i}{n}
\]
Where \( d_i \) are the deviations from the assumed value \( C \).
Example:
Let’s use the same observations and choose \( C = 50 \):
Deviations:
\( d_1 = 54 – 50 = 4 \)
\( d_2 = 56 – 50 = 6 \)
\( d_3 = 70 – 50 = 20 \)
\( d_4 = 45 – 50 = -5 \)
\( d_5 = 50 – 50 = 0 \)
Now, applying the shortcut method:
\[
\bar{X}_{ar} = 50 + \frac{(4 + 6 + 20 – 5 + 0)}{5} = 50 + \frac{25}{5} = 50 + 5 = 55
\]
Video
Conclusion
As you can see, both methods yield the same result: the arithmetic mean is 55. In upcoming videos, I’ll explain the arithmetic mean for different types of data and delve into the other measures of central tendency.
Thanks for watching! I hope you found this explanation helpful. Stay tuned for more insights on statistics!