5.2.1.2.1. Discrete Series

In case of discrete series, frequency against each observation is multiplied by the value of the observation. The values, so obtained, are summed up and divided by the total number of frequencies. Symbolically, $X = \frac{\Sigma fX}{\Sigma f}$ Where, $\Sigma fX = \text{sum of the product of variables and frequencies.}$ $\Sigma f = \text{sum of frequencies.}$

In the Direct Method of calculating the arithmetic mean for a discrete series, you follow these steps:

1. Multiply the frequency of each observation by the value of that observation (this gives you $fX$).
2. Sum all of the products you created in the first step to get $\Sigma fX$.
3. Sum all the frequencies to get $\Sigma f$.
4. Divide $\Sigma fX$ by $\Sigma f$ to get the arithmetic mean, $X$. This method is straightforward as it directly uses the frequencies tied to each observation.

Think of a simple example where you have three types of fruit in a basket: 100 apples, 50 bananas, and 10 oranges. If you wanted to find the average weight of the fruit in your basket, you would take the weight of each type of fruit, multiply that by how many you have (that's like the frequency), and then divide it by the total number of pieces of fruit to get an 'average weight' per piece.

Example: Plots in a housing colony come in only three sizes: 100 sq. metre, 200 sq. meters, and 300 sq. metre, and the number of plots are respectively 200, 50, and 10. $\text{Arithmetic mean using direct method, } X = \frac{33000}{260} = 126.92 \text{ Sq. metre}$. Therefore, the mean plot size in the housing colony is 126.92 Sq. metre.

In the example provided:

1. You identify the sizes of the plots and their respective frequencies (number of plots).
2. Compute the total of the products of each plot size and its corresponding frequency, which gives you $33000$.
3. Then add the total number of plots, which gives you $260$.
4. Finally, divide $33000$ by $260$ to find the average plot size. This method clearly demonstrates how to compute a mean based on grouped data.

Imagine you're trying to find the average size of a group of friends' gardens, where most have small gardens (100 sq m), a few have medium (200 sq m), and a couple have large gardens (300 sq m). By calculating the contributions of each garden size based on how many friends have that size, you'll easily find the average garden size of your group.

As in case of individual series, the calculations can be simplified by using assumed mean method, as described earlier, with a simple modification. Since frequency (f) of each item is given here, we multiply each deviation (d) by the frequency to get fd. Then we get $\Sigma fd$. The next step is to get the total of all frequencies i.e. $\Sigma f$. Finally, find out $\frac{\Sigma fd}{\Sigma f}$. The arithmetic mean is calculated by $X = A + \frac{\Sigma fd}{\Sigma f}$ using assumed mean method.

The Assumed Mean Method involves the following steps:

1. Choose an assumed mean value (A) for your dataset.
2. Compute the deviation (d) from this assumed mean for each observation.
3. Multiply these deviations by their respective frequencies to obtain $fd$.
4. Sum up all the results to find $\Sigma fd$ and also sum the frequencies to get $\Sigma f$.
5. Finally, use the formula to calculate the actual mean by adding the assumed mean (A) to the ratio of $\Sigma fd$ to $\Sigma f$. This method reduces the complexity of direct calculations.

Think of choosing a midpoint value for the amount of time students study each day. If you assume they roughly study 2 hours per day, you can then calculate how much more or less other students study compared to this 'average.' By aggregating these deviations and accounting for how many students study that amount of time, you can quickly adjust your estimated average study time.

In this case, the deviations are divided by the common factor ‘c’ which simplifies the calculation. Here we estimate $d' = \frac{d}{c}$. Then get $fd'$ and $\Sigma fd'$. The formula for arithmetic mean using step deviation method is given as, $X = A + c \frac{\Sigma fd'}{\Sigma f}$.

The Step Deviation Method is useful when the calculation becomes tedious due to large numbers. Here's how it works:

1. Select a common factor (c) which simplifies the calculations, often a rounding number.
2. Replace your deviations (d) with their simpler forms (d') by dividing by the common factor.
3. Multiply the new deviations by respective frequencies to get $fd'$ and sum them up to find $\Sigma fd'$.
4. Finally, substitute everything into the step deviation formula to find your mean. This method makes calculations easier and helps avoid errors in arithmetic.

Imagine you are measuring heights in centimeters, and the numbers are large (like 172, 180, etc.). Instead, you could convert these heights into a simpler scale by rounding them down (like taking out the hundreds) to make calculations easier — for instance, using just 72, 80, etc. This allows you to handle the data quickly while still maintaining accuracy in the final result.

• Find the mean plot size for the data given in example 3, by using step deviation and assumed mean methods.

This activity task challenges you to apply what you've learned. Using the data from example 3:

1. First, try to calculate the mean plot size using both the Step Deviation Method and the Assumed Mean Method.
2. This will involve setting up your assumed mean (for the Assumed Mean Method) and choosing a common factor (for the Step Deviation Method) to use in your calculations.
3. The goal is to reinforce understanding through practical application of the methods discussed.

Think of it like playing a game where you get to practice your skills. In this case, you’re not just reading about how to calculate the mean; you’re actually playing out the game with real numbers and applying the strategies you've just learned!

--