Estimating the Population Mean

• The population mean is the average of all values in an entire population.
• It can be estimated by using a sample mean, which is the average of a subset of the population.
• To calculate the sample mean, add all the values together and divide by the number of values. This is represented as Σx/n, where x represents the values and n is the number of values.
• Since the mean is a single number representing the entire dataset, it can sometimes be influenced by outliers. Extreme values may cause the mean to be skewed.
• The sample mean is a point estimate of the population mean. It is a single number estimate, but does not account for uncertainty.
• A more comprehensive estimate includes an interval estimate which provides a range of values likely to contain the population mean, usually together with a level of confidence.

Estimating the Population Standard Deviation

• Standard deviation is a measure of spread in the data. It measures how much individual data points differ from the mean.
• Similar to the mean, we can estimate the population standard deviation by calculating the sample standard deviation.
• To calculate the sample standard deviation, take each value in the sample, subtract the sample mean, square the result, and then work out the average of these squared differences. This is known as the mean square deviation or variance.
• The standard deviation is then found by taking the square root of the variance.
• It is important to note that like the mean, the standard deviation can also be influenced by outliers.
• A smaller standard deviation indicates that the values are closer to the mean, while a larger standard deviation indicates that the values are spread out over a greater range.

Using Estimates for Predictions

• Knowledge of the sample mean and standard deviation can be used to make predictions about the population.
• A commonly used prediction is the range rule of thumb which states that almost all values will fall within the range of mean ± 2 standard deviations.
• However, for more precise predictions or when the distribution of data is not known, other statistical methods such as confidence intervals and hypothesis tests can be utilised.
• The use of sample-based estimates is an essential part of inferential statistics, which seeks to make predictions and draw conclusions about a population based on a sample.
• Remember that these are all estimates, and true population parameters are usually unknown. Each sample may yield a slightly different estimate, due to sampling variability.