91Ó°ÊÓ

Calculating the mean is one of the foundational steps in statistical analysis. It offers a simple average to summarize a dataset in one number and helps to provide a sense of the data's central point. In order to calculate the mean, you need to find the sum of all data points and divide it by the count of data points.

For example, with weights given as

• 51 kg
• 69 kg
• 57 kg
• 61 kg
• 75 kg
• 105 kg etc.

we first add the weights: \[51 + 69 + 69 + 57 + 61 + 57 + 75 + 105 + 69 + 63 = 676\]Next, we count the weights, which is 10. Finally, we divide the total sum by the count:\[\text{Mean} = \frac{676}{10} = 67.6 \, \text{kg}\]Thus, the mean weight of the students is 67.6 kilograms, offering a simple yet informative glimpse into the central value of the dataset.

Variance measures how much a set of numbers differs from their mean. It takes into account the squared differences and provides a sense of the data's spread. Calculating variance involves several steps.

• First, determine each number's deviation from the mean.
• Then, square each deviation to ensure they are positive.

This results in squared deviations:

Examples include:

• For a weight of 51 kg: deviation is \(51 - 67.6 = -16.6\), squared gives \((-16.6)^2 = 275.56\).
• For another, 105 kg: deviation is \(105 - 67.6 = 37.4\), squared gives \((37.4)^2 = 1397.76\).

After calculating each squared deviation, sum these values: \[275.56 + 1.96 + 1.96 + 112.36 + 43.56 + 112.36 + 54.76 + 1397.76 + 1.96 + 21.16 = 2023.4\]Finally, divide the total by the count of numbers to get variance:\[\text{Variance} = \frac{2023.4}{10} = 202.34\]This number shows that the weights are moderately spread out from the mean.

Standard deviation provides insights into the amount of variation or dispersion from the average. It's the square root of the variance and indicates how spread out the numbers are around the mean.

To calculate, we simply take

• the square root of the variance calculated in the previous step.\[\sqrt{202.34} \approx 14.22\]

Thus, the standard deviation is approximately 14.22 kg. This tells us about the average distance of the weights from the mean weight. A smaller standard deviation would indicate that the weights are clustered closely around the mean, while a larger one suggests they are more spread out.