5. Application to Grouped Data

For grouped frequency data, use:

$\bar{x} = \frac{\sum f x}{\sum f}$ Where: • 𝑓 is the frequency of the class, • 𝑥 is the mid-point of each class.

To calculate the mean for grouped data, we utilize the midpoints of different data intervals (or classes) and their associated frequencies. The formula $\bar{x} = \frac{\sum f x}{\sum f}$ indicates that we multiply each class midpoint by its frequency, sum these products, and then divide by the total number of observations represented by the total frequency. This method helps us simplify the calculation when dealing with ranges of data instead of individual data points.

Imagine you are trying to determine the average height of students in a school where heights are grouped into intervals, like 140-150 cm, 150-160 cm, etc. Each height range has a certain number of students (frequency). Instead of measuring the height of each student, you take the average height of each range (midpoint) and calculate the overall average using the numbers of students in each range.

To find the standard deviation for grouped data:

$\sigma = \sqrt{\frac{\sum f (x - \bar{x})^2}{\sum f}}$ Where: • 𝑓 is the frequency of the class, • 𝑥 is the mid-point of each class.

To compute the standard deviation for grouped data, we first calculate the mean as mentioned earlier. Next, we find the deviation of each class midpoint from the mean, square this deviation, and multiply by the frequency of that class. We sum all these values, divide by the total frequency, and finally take the square root of this result to obtain the standard deviation. This gives us an understanding of how the data is distributed around the mean.

Consider a classroom where students' test scores are grouped into ranges, such as 0-50, 51-100, etc. After finding the average score, we then assess how much scores vary from this average, giving more importance to higher discrepancies. This helps to identify if most students scored closely around the average or if some scored much lower or higher, providing insights into the overall performance.

Example 2: Grouped Data Marks (Class Interval) Frequency 0 – 10 2 10 – 20 3 20 – 30 5 Step 1: Find midpoints • 5, 15, 25 Step 2: Multiply by frequency (fx) • 2×5 = 10, 3×15 = 45, 5×25 = 125 • ∑𝑓𝑥 = 180, ∑𝑓 = 10 Mean: $\bar{x} = \frac{180}{10} = 18$ Step 3: Find 𝑓(x - mean)²

In this example, we're finding the mean and standard deviation for test scores grouped in intervals. We first identify the midpoints of each interval. Then, we multiply these midpoints by their respective frequencies to get a total for all groups combined. Next, we divide this sum by the total frequency to find the mean. After that, we calculate the squared deviations from the mean for each midpoint, weighted by their frequency, allowing us to estimate the spread of our grouped data.

Suppose you have a bakery and you keep track of how many pastries you sell in different price ranges, such as $0-$5, $5-$10. By calculating the midpoints of these ranges, then multiplying by how many pastries were sold in each range, you can find the average selling price of pastries. This allows you to gauge how well your pricing strategy is working and if adjustments are needed based on the sales spread.

--