4. Step-by-Step Example

Consider the marks out of 10: 3,5,7,5,10

In this example, we have a sample of marks scored by students out of 10. The values are 3, 5, 7, 5, and 10. We will perform a series of calculations to understand how to derive the mean, variance, and standard deviation from this data set.

Imagine a teacher evaluating her class. She wants to know how her students performed on a test scored out of 10 and will analyze these scores to gauge class performance.

3+5+7+5+10 30 𝑥‾ = = = 6 5 5

To find the mean, we sum all the marks (3 + 5 + 7 + 5 + 10 = 30) and then divide by the number of students (5). Thus, the mean (average) score is 30 divided by 5, which equals 6.

Think of the mean as the average score on a sports team; it gives a quick overview of how the team performs as a whole.

𝑥 𝑥 −𝑥‾ (𝑥 −𝑥‾)² 𝑖 𝑖 𝑖 3 -3 9 5 -1 1 7 1 1 5 -1 1 10 4 16

We calculate how much each individual mark deviates from the mean (which we found to be 6). We subtract the mean from each score to find the deviation. Next, we square each deviation to ensure they are positive. For instance, for the score of 3, the deviation is 3 - 6 = -3; squaring this gives us 9.

Like measuring how far each student is from the average seat height in a classroom; squaring their heights emphasizes those who are much shorter or taller than average.

9+1+1+1+16 28 𝑠² = = = 7 5−1 4

We sum the squared deviations calculated in the previous step (9 + 1 + 1 + 1 + 16 = 28). To find the variance (denoted by s² for a sample), we need to divide this sum by the total number of values minus one (n - 1), which in this case is 4. Thus, 28 divided by 4 equals 7.

Similar to understanding how widely students' heights differ in a class; variance provides an average measure of this spread.

𝑠 = √7 ≈ 2.65

Finally, the standard deviation (s) is found by taking the square root of the variance. So, we find the square root of 7, which is approximately 2.65. This value represents the typical distance of the marks from the mean.

Think about measuring how unevenly the students performed on the test; a higher standard deviation indicates that some students scored significantly higher or lower than the average.

For grouped frequency data, use: ∑𝑓𝑥 ∑𝑓(𝑥−𝑥‾)² 𝑥‾ = 𝜎 = √ ∑𝑓 ∑𝑓 Where: • 𝑓 is the frequency of the class, • 𝑥 is the mid-point of each class.

When we have grouped data, we need to adjust our calculations. We find the midpoint of each class interval and then multiply these midpoints by their corresponding frequencies (how many data points fall in each interval). This allows us to find the mean and subsequently compute variance and standard deviation.

Imagine a survey asking people about their age ranges. Instead of individual ages, we get groups (e.g., 0-10, 10-20, etc.). We calculate measures based on these groups to get an overall understanding of age distribution.

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

In this example, we have grouped data for marks within specific ranges. The first step is to find the midpoints of each interval (e.g., the midpoint of 0-10 is 5, of 10-20 is 15, etc.). Following that, we calculate the total by multiplying each midpoint by the frequency of that interval to get the weighted summary of scores.

Imagine collecting data on the annual income of households. Instead of exact incomes, we look at ranges. We analyze these ranges to understand how wealth is distributed across the population.

Mean: 180 𝑥‾ = = 18 10 Step 3: Find 𝑓(𝑥 −𝑥‾)² x f x - mean (x - mean)² f(x - mean)² 5 2 -13 169 338 15 3 -3 9 27 25 5 7 49 245 338+27+245 610 𝜎 = √ = √ = √61 ≈ 7.81 10 10

We continue the grouped data analysis by calculating f(x - mean)² using the midpoints. We find the deviation of the midpoints from the computed mean of 18 and square it. Finally, we sum these squared values to determine the variance and compute the standard deviation by taking the square root.

Like evaluating the performance of different retail stores in a mall; we take note of how much each store's revenue deviates from the average revenue to gauge overall performance variability.

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