# Frequency Distributions and Measures of Central Tendency

Describe frequency distributions and measures of central tendency and dispersion

## Frequency Distributions

Frequency distributions are a method of tabulating or graphically displaying a number of observations.

### The Normal Distribution

The normal distribution is a gaussian distribution, where the majority of values cluster around the mean, and whilst more extreme values become progressively less frequent.

The normal distribution is common in medicine for two reasons.

• Much of the variation in biology follows a normal distribution
• When multiple random samples are taken from a population, the mean of these samples follows a normal distribution, even if the characteristic being measured is not normally distributed
This is known as the central limit theorem.
• It is useful because many statistical tests are only valid when the data follow a normal distribution

The formula for the normal distribution is given by:

From this, it can bet seen the two variables which will determine the shape of the normal distribution are:

• μ (mu): The mean
• σ (sigma): The standard deviation

#### The Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The equation for the standard normal distribution is much simpler, which is why it is used.

Any normal distribution can be transformed to fit a standard normal distribution using a z transformation:

The value of z then gives a standardised score, i.e. the number of standard deviations form the mean in a standardised curve. This can then be used to determine probability.

### Binomial distribution

Where observations belong to one of two mutually exclusive categories, i.e.:
If then

If the number of observations is very large and the probability of an event is small, a Poisson distribution can be used to approximate a binomial distribution.

## Measures of Central Tendency

As noted above in the normal distribution, results tend to cluster around a central value. Quantification of the degree of clustering can be done using measures of central tendency, of which there are three:

• Mode
The most common value in the sample.
• Median
The middle value when the sample is ranked from lowest to highest.
• The median is the best measure of central tendency when the data is skewed
• Arithmetic mean
The average, i.e:
The mean is common and reliable, though inaccurate if the distribution is skewed.

## Measures of Dispersion

Measures of variability describe the degree of dispersion around the central value.

### Basic Measures of Deviation

• Range: The lowest and highest values in the sample
Highly influenced by outliers
• Percentiles: Rank observations into 100 equal parts, so that the median becomes the 50% percentile.
Better measure of spread than range.
• Interquartile range: The 25th to 75th centile
A box-and-whisker plot graphically demonstrates the mean, 25th centile, 75th centile, and (usually), the 10th and 90th centiles.
• Outliers are represented by dots
• Occasionally the range is plotted by the whiskers, and there are no outliers plotted

### Variance and Standard Deviation

Variance is a better measure of variability than the above methods. Variance:

• Evaluates how far each observation is from the mean, and penalises observations more the further they lie from the mean
• Sums the squares of each difference and divides by the number of observations i.e:
• is used (instead of ) because the mean of the sample is known and therefore the last observation calculated must taken on a known quantity
• This is known as a degrees of freedom, which is a mathematical restriction used when using one statistical test in order to estimate another
• It is a confusing topic best illustrated with an example:
• You have been given a sample of two observations (say, ages of two individuals), and you know nothing about them
• The degrees of freedom is two, since those observations can take on any value.
• Alternatively, imagine you have been given the same sample, but this time I tell you that the mean age of the sample is 20
• The degrees of freedom is one, since if I tell you the value of one of the observations is 30, you know that the other must be 10
Therefore, only one of the observations is free to vary - as soon as its value is known then the value of the other observation is known as well.
• Different statistical tests may result in additional losses in degrees of freedom.

#### Standard Deviation

The standard deviation is the positive square root of the variance.

In a sample of normal distribution:

• 1 SD either side of the mean should include ~68% of results
• 2 SD either side of the mean should include ~95% of results
• 3 SD either side of the mean should include ~99.7% of results

### Standard error and Confidence Intervals

Standard error of the mean is:

• A measure of the precision of the estimate of the mean
• Calculated from the standard deviation and the sample size
As the sample size grows, the SEM decreases (as the estimate becomes more precise).
• Given by the formula:
• Used to calculate the confidence interval

#### Confidence Interval

The confidence interval:

• Gives a range in which the true population parameter is likely to lie
The width of the interval is related to the standard error, and the degree of confidence (typically 95%):
• Is a function of the sample statistic (in this case the mean), rather than the actual observations
• Has several benefits over the p-value:
• Indicates magnitude of the difference in a meaningful way
• Indicates the precision of the estimate
The smaller the confidence interval, the more precise the estimate.
• Allows statistical significance to be calculated
If the confidence interval crosses 1, then the result is insignificant.

## References

1. "Normal distribution". Licensed under Attribution 3.0 Unported (CC BY 3.0) via SubSurfWiki.
2. Myles PS, Gin T. Statistical methods for anaesthesia and intensive care. 1st ed. Oxford: Butterworth-Heinemann, 2001
3. Course notes from "Introduction to Biostats", University of Sydney, School of Public Health, circa 2013.
Last updated 2019-07-18