Q.The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results: Number of observations = 25, mean = 18.2 seconds, standard deviation = 3.25 seconds. Further, another set of 15 observations $x_1, x_2, \ldots, x_{15}$, also in seconds, is now available and we have $\sum_{i=1}^{15} x_i = 279$ and $\sum_{i=1}^{15} x_i^2 = 5524$. Calculate the standard deviation based on all 40 observations.

The problem requires us to combine two sets of data and calculate the standard deviation for the entire combined set. This involves finding the total sum of observations and the total sum of squares for all data points.

Concept: Combined Mean and Standard Deviation

1. First, we find the sum of observations ($\sum x_1$) and the sum of squares ($\sum x_1^2$) for the initial 25 observations.

   Given $n_1 = 25$, $\bar{x}_1 = 18.2$, $\sigma_1 = 3.25$.

   $\sum x_1 = n_1 \bar{x}_1 = 25 \times 18.2 = 455$.

   Using the formula $\sigma_1^2 = \frac{\sum x_1^2}{n_1} - \bar{x}_1^2$, we get $\sum x_1^2 = n_1 (\sigma_1^2 + \bar{x}_1^2)$.

   $\sum x_1^2 = 25 (3.25^2 + 18.2^2) = 25 (10.5625 + 331.24) = 25 (341.8025) = 8545.0625$.

2. Next, we combine these sums with the given sums for the second set of 15 observations.

   Total number of observations $N = 25 + 15 = 40$.

   Total sum of observations $\sum X = 455 + 279 = 734$.

   Total sum of squares $\sum X^2 = 8545.0625 + 5524 = 14069.0625$. …

When combining two sets of data to find a new mean or standard deviation, we cannot simply average the individual means or standard deviations. This is because the mean is a sum divided by count, and the standard deviation (or variance) involves sums of squares. These underlying sums are the fundamental quantities that can be directly combined.

The mean of a dataset is defined as $\bar{x} = \frac{\sum x}{n}$, where $\sum x$ is the sum of all observations and $n$ is the number of observations.

The variance, $\sigma^2$, is defined as $\sigma^2 = \frac{\sum x^2}{n} - (\bar{x})^2$, where $\sum x^2$ is the sum of the squares of all observations.

From these definitions, we can derive the sum of observations and the sum of squares:

$\sum x = n \bar{x}$

$\sum x^2 = n(\sigma^2 + \bar{x}^2)$

These derived sums are what we need to combine. Once we have the total sum of observations and the total sum of squares for the combined dataset, we can then calculate the overall mean and variance, and finally the standard deviation.

Here's how we approach the problem:

1. Extract information and calculate sums for the first set of observations.

   We are given:

   • Number of observations, $n_1 = 25$
   • Mean, $\bar{x}_1 = 18.2$ seconds
   • Standard deviation, $\sigma_1 = 3.25$ seconds

   First, we find the sum of observations for this set:

Next, we find the sum of squares for this set. We know that $\sigma_1^2 = \frac{\sum x_1^2}{n_1} - (\bar{x}_1)^2$.
Rearranging this formula to solve for $\sum x_1^2$:

Let's calculate $\sigma_1^2$ and $\bar{x}_1^2$:

Now, substitute these values into the formula for $\sum x_1^2$:

2. Extract information for the second set of observations.

   We are given:

   • Number of observations, $n_2 = 15$
   • Sum of observations, $\sum x_2 = 279$
   • Sum of squares of observations, $\sum x_2^2 = 5524$

   For this set, the necessary sums are already provided, so no further calculations are needed here.

3. Combine the data to find the total number of observations, total sum, and total sum of squares.

   Let $N$ be the total number of observations, $\sum X$ be the total sum of observations, and $\sum X^2$ be the total sum of squares for the combined data.

4. Calculate the mean of the combined data. …