In Problems 5–10, by hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated.
Population: 1, 19, 25, 15, 12, 16, 28, 13, 6

Verified step by step guidance

1. Identify that the data given represents the entire population, so we will calculate the population variance and population standard deviation.
2. Calculate the population mean (\( \mu \)) using the formula: \[\mu = \frac{\sum_{i=1}^N x_i}{N}\] where \( N \) is the number of data points and \( x_i \) are the individual data values.
3. Find the squared differences from the mean for each data point: \[ (x_i - \mu)^2 \] Calculate this for each \( x_i \) in the population.
4. Calculate the population variance (\( \sigma^2 \)) using the formula: \[\sigma^2 = \frac{\sum_{i=1}^N (x_i - \mu)^2}{N}\] This is the average of the squared differences.
5. Find the population standard deviation (\( \sigma \)) by taking the square root of the variance: \[\sigma = \sqrt{\sigma^2}\]

Key Concepts

Population Variance

Population variance measures the average squared deviation of each data point from the population mean. It quantifies how spread out the entire population data is. To calculate it, find the mean of all values, subtract the mean from each value, square the result, sum these squares, and divide by the total number of data points.

Population Standard Deviation

Population standard deviation is the square root of the population variance. It expresses the spread of data in the same units as the original data, making it easier to interpret variability. A smaller standard deviation indicates data points are closer to the mean, while a larger one shows more dispersion.

Difference Between Population and Sample Variance

Population variance uses all data points and divides by N, the total number of values. Sample variance estimates population variance from a subset and divides by (n-1) to correct bias. Understanding this distinction is crucial for choosing the correct formula based on whether data represents the entire population or a sample.