Question

When a researcher selects all possible pairs of samples from a population in order to find the difference between the means of each pair, what will be the shape of the distribution of the differences when the original distributions are normally distributed? What will be the mean of the distribution? What will be the standard deviation of the distribution?

Short answer

The distribution is normal with mean \( \mu = 0 \) and standard deviation \( \sigma_{difference} = \sqrt{2} \times \sigma \).

Step-by-step solution

Understanding the Problem

We need to find the distribution of the differences of means from pairs of samples from a normally distributed population. We're asked about the shape, mean, and standard deviation of this distribution.

Determine the Shape of the Distribution

If the original population is normally distributed, the distribution of the differences of means for all possible pairs will also be normally distributed. This is because the difference of two normally distributed variables is also normally distributed.

Calculate the Mean of Differences

The mean of the distribution of differences of means remains the same as the mean of the population from which the samples were drawn. This is because each sample mean is an unbiased estimator of the population mean, so the mean of differences is zero, \( \mu = 0 \).

Determine the Standard Deviation of Differences

The standard deviation of the differences (\( \sigma_{difference} \)) can be found using the standard deviation of the original distribution (\( \sigma \)). Since each sample pair is independent, and assuming equal variances, the standard deviation of the differences is given by \( \sigma_{difference} = \sqrt{2} \times \sigma \).

Key concepts

Sampling Distribution

The term *sampling distribution* refers to the probability distribution of a statistic obtained by selecting random samples from a population. When we talk about sampling distributions, we're usually focused on understanding how a sample statistic, such as the mean or variance, behaves across multiple samples.
If you collect multiple samples from the same population and calculate the desired statistic for each sample, the distribution of these statistic values will form the sampling distribution.

• For instance, consider collecting a series of random samples and calculating the mean for each. The distribution of these means is a sampling distribution known as the sampling distribution of the sample mean.
• The sampling distribution helps us estimate the population parameter, such as the population mean, more accurately since it considers the variability between different samples.

Understanding sampling distribution is fundamental in inferential statistics because it forms the basis for estimating population parameters and testing hypotheses.

Normal Distribution

The *normal distribution* is one of the most important concepts in statistics. It's often called the Gaussian distribution and is characterized by its perfectly symmetrical bell-shaped curve. In a normal distribution, most of the data points are around the mean, with fewer and fewer falling further away as you move towards the tail ends.

• In a perfectly normal distribution, the mean, median, and mode are identical and sit at the center of the curve.
• The shape of the normal distribution is completely defined by its mean and standard deviation, with the mean determining the center of the distribution and the standard deviation describing the spread.

If the data you are working with is normally distributed, many statistical methods apply directly, making the analysis simpler. Furthermore, even if the data isn't exactly normally distributed, the Central Limit Theorem tells us that the means of a sufficiently large number of samples from this population will approximate a normal distribution, which is very useful for inferential statistics.

Standard Deviation

The *standard deviation* is a measure that describes the amount of variation or dispersion in a set of values. A low standard deviation means that the values are generally close to the mean, indicating consistency, while a high standard deviation means that the data points are spread out over a wider range.

• The formula for calculating the standard deviation of a sample is given by \[ s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n}(x_i - \bar{x})^2}\]where \( x_i \) represents each data point in your sample, \( \bar{x} \) is the sample mean, and \( n \) is the number of observations.
• Standard deviation is crucial in the context of normal distribution because it allows us to understand how spread out the data points are relative to the mean. Approximately 68% of data in a normal distribution lies within one standard deviation of the mean, 95% within two, and 99.7% within three.

Understanding standard deviation helps in making predictions about the spread of values in a data set and is vital when comparing variability between different data sets.

Population Mean

The *population mean* is the average of all the data points in a population, offering a central value around which the rest of the data is distributed. In statistics, it is commonly denoted as \( \mu \) and represents the true mean of the entire dataset from which your sample may be drawn.

• The population mean is calculated using the formula: \[ \mu = \frac{1}{N} \sum_{i=1}^{N} x_i\] where \( N \) is the total number of data points, and \( x_i \) are each value in the population.
• While it is an ideal parameter, often we must estimate it using a sample mean when it is not feasible to measure the entire population. This estimation is where sampling distribution principles are critical. By taking many samples and calculating their means, we can get an accurate estimate of \( \mu \).

Understanding how the population mean is estimated in practice is essential for statistical analysis and for drawing conclusions about the larger data set from which samples are taken.