Q.Find the variance and standard deviation of the following scores on an exam: 92, 95, 85, 80, 75, 50.

To find the variance and standard deviation of a set of scores, we first need to calculate the mean of the scores.

1. Calculate the Mean ($\mu$): The mean is the sum of all scores divided by the number of scores. Given scores: $92, 95, 85, 80, 75, 50$. Number of scores ($N$) = $6$.

2. Calculate the Variance ($\sigma^2$):

   Variance measures the average of the squared differences from the mean.

   $\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$

   Where $x_i$ are the individual scores, $\mu$ is the mean, and $N$ is the number of scores.

   Let's calculate $(x_i - \mu)^2$ for each score:

   • $(92 - 79.5)^2 = (12.5)^2 = 156.25$
   • $(95 - 79.5)^2 = (15.5)^2 = 240.25$
   • $(85 - 79.5)^2 = (5.5)^2 = 30.25$
   • $(80 - 79.5)^2 = (0.5)^2 = 0.25$
   • $(75 - 79.5)^2 = (-4.5)^2 = 20.25$
   • $(50 - 79.5)^2 = (-29.5)^2 = 870.25$

   Sum of squared differences:

Now, calculate the variance:
$$ \sigma^2 = \frac{1317.5}{6} \approx 219.5833 $$ …

When analyzing a set of data, such as exam scores, we often want to understand not just the average score but also how spread out the scores are. Measures of central tendency (like the mean) tell us about the typical value, but they don't tell us if all scores are close to the average or if they vary widely. This is where measures of dispersion come in.

Variance and Standard Deviation are two fundamental measures of dispersion. They quantify the extent to which individual data points in a set deviate from the mean of the set.

Understanding Variance ($\sigma^2$)

Variance measures the average of the squared differences from the mean.

• Why squared differences? If we just summed the differences from the mean, the positive and negative deviations would cancel out, always resulting in zero. Squaring the differences ensures all values are positive, so they contribute to the total spread.
• What does it tell us? A high variance indicates that data points are generally far from the mean, and far from each other. A low variance indicates that data points are generally close to the mean.
• Units: The unit of variance is the square of the original data's unit (e.g., if scores are in marks, variance is in marks squared), which can make it difficult to interpret directly.

Understanding Standard Deviation ($\sigma$)

Standard deviation is the square root of the variance.

• Why take the square root? Taking the square root brings the measure of dispersion back to the original units of the data. This makes it much easier to interpret than variance.
• What does it tell us? It tells us, on average, how much each score deviates from the mean. A small standard deviation means scores are clustered closely around the mean, while a large standard deviation means scores are more spread out.

Step-by-Step Calculation

Let's calculate the variance and standard deviation for the given exam scores: 92, 95, 85, 80, 75, 50.

Step 1: Calculate the Mean ($\mu$)

The mean is the sum of all scores divided by the number of scores.

Scores ($x_i$): 92, 95, 85, 80, 75, 50

Number of scores ($N$) = 6

Step 2: Calculate the Deviation of Each Score from the Mean ($x_i - \mu$)

Subtract the mean (79.5) from each score.