"Population standard deviation"

Standard deviation is a measure of the dispersion or spread of a set of data points relative to their mean. For populations, the standard deviation is denoted by \sigma.

Population vs Sample: The population refers to the entire group being studied, while a sample is a subset of that population. The calculation of the standard deviation differs for populations and samples.
Mean (\mu): The average of the values in the population. It is calculated using the formula:
\mu = \frac{1}{n} \sum{i=1}^{n} xi where x_i are the data points and n is the total number of points.

Calculate the Mean (\mu):
- If the weights lost are represented as x1, x2, … , xn, the mean is: \mu = \frac{x1 + x2 + … + xn}{n}
Find the Squared Deviations:
- Calculate the squared differences from the mean for each data point:
  - (x1 - \mu)^2, (x2 - \mu)^2, …, (x_n - \mu)^2
Calculate the Variance (\sigma^2):
- The population variance is the average of the squared deviations:
  \sigma^2 = \frac{1}{N} \sum{i=1}^{N} (xi - \mu)^2 where N is the number of data points.
Determine the Population Standard Deviation (\sigma):
- Finally, the standard deviation is the square root of the variance:
  \sigma = \sqrt{\sigma^2}

Given Weights Lost: Assuming the weights lost are 5, 4, 4, 5, 2.
Calculate the Mean:
\mu = \frac{5 + 4 + 4 + 5 + 2}{5} = 4
Calculate Squared Deviations:
- For each weights lost:
- (5 - 4)^2 = 1
- (4 - 4)^2 = 0
- (4 - 4)^2 = 0
- (5 - 4)^2 = 1
- (2 - 4)^2 = 4
Variance Calculation:
- \sigma^2 = \frac{1 + 0 + 0 + 1 + 4}{5} = \frac{6}{5} = 1.2
Standard Deviation:
- \sigma = \sqrt{1.2} ≈ 1.095
- Rounded to two decimal places: \sigma ≈ 1.10

Population Standard Deviation Formula:
- The general formula is:
  \sigma = \sqrt{\frac{\sum{i=1}^{N} (xi - \mu)^2}{N}}
The standard deviation provides important insights into the variability of data and is crucial in statistical analysis.