Chapter 2 Summary Statistics
A statistic is the generic name given to any quantity calculated from the data. A statistic is therefore a function of the data.
We use a summary statistic to measure and summarise some aspect of the data. Many simple summary statistics can be divided as one of two types: measures of location (Section 2.1), and measures of spread (Section 2.2).
A measure of location is a value “typical” of the data. Examples include the mean, mode and median.
A measure of spread quantifies how variable the data are. Examples include the variance, standard deviation, range and interquartile range.
2.1 Measures of location
A statistic that is a measure of location summarises the data by giving a single number that is, in some sense, typical of that set of data. There are three commonly used measures of location, of which the mean is the most important.
- Mean Suppose that \(n\) measurements have been taken on the
random variable under investigation, and these are denoted by
\(x_1 , x_2 , \ldots , x_n\). The (arithmetic) mean of the
observations is usually denoted by \(\bar{x}\) and is given by
\[\begin{equation} \bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{1}{n} \sum\limits_{i=1}^n x_i. \tag{2.1} \end{equation}\] In everyday language we say that \(\bar {x}\) is the average of the observations.
For an explanation of (2.2) and derivation of the mean of the optometry students data, see Video 2.
Video 2: Derivation of the mean
Sometimes continuous data are grouped into a frequency table — this loses the values of the individual and essentially converts the continuous data to discrete data. The sample mean in this case be estimated by assuming that all the observations in a class interval fall at the mid-point of the interval. Therefore, if we have \(k\) intervals with mid-points \(m_1\), \(m_2\), \(\dots\), \(m_k\) and frequencies \(f_1 , f_2 , \ldots , f_k\) then we treat the mid-points as though they are the discrete data values and we can use the formula in (2.2). When determining the midpoint of an interval, due notice should be taken of any rounding of the individual observations.
- Median. The sample median is defined as the observation with cumulative frequency of \(n/2\). This is sometimes used instead of the mean, particularly when the histogram of the observations looks asymmetric. The median is obtained by sorting the observations in ascending order and then picking out the middle observation or, equivalently, the \(\frac{(n+1)}{2}\)th observation. If there are an even number of observations in the sample then the median is taken as the average of the \(\frac{n}{2}\)th and \(\left(\frac{n}{2}+1\right)\)th observations. The ranking can be done using a stem-and-leaf plot.
Hence the median is the ‘middle value’ in the sample with half the observations numerically greater than the median and the other half smaller than the median. The mathematical properties of the median are less easy to determine than those of the mean and so the mean is preferred for use in most formal statistical analyses. However, the median is quick to calculate in small samples and is useful in descriptive work.
- Mode. This is the value of the random variable which occurs with greatest frequency. The sample mode for discrete data can be found easily by inspection of the data or by a quick look at a frequency table. It is not realistic to define the sample mode for continuous data (we will see why when introducing continuous probability distributions in Section 5). However when such data have been classified into intervals or categories we can find the modal class — the class or interval containing the most number of observations.
2.2 Measures of spread
Given the location of the data, we next need to know whether the observations are concentrated around this value, or dispersed over a much wider range of values. Measures of spread express this idea numerically.
- Variance. The sample variance of \(n\) observations,
\(x_1 , x_2 , \ldots, x_n\), is usually denoted \(s^2\) and defined as
\[\begin{equation} \begin{aligned} s^2 &= \frac{(x_1 - {\bar{x}})^2 + (x_2 - {\bar{x}})^2 + \cdots + (x_n - {\bar{x}})^2}{n-1}\\ &= \frac{1}{n-1} \sum\limits_{i=1}^{n} (x_i - {\bar{x}})^2.\end{aligned} \tag{2.3} \end{equation}\] It is an average squared distance of the observations from their sample mean value. The adjustment is that the divisor is \((n-1)\) rather than \(n\) to correct for a bias (we’ll come back to this later in Section 9.3) which occurs because we are measuring from the mean of the data rather than the true mean of the population. It is worth pointing out that the difference between using \(n\) and \((n-1)\) is especially important for small samples. This formula can be rearranged to the alternative form
\[\begin{equation} \begin{aligned} s^2 &= \frac{1}{n-1} \left( \sum\limits_{i=1}^{n} x_i^2 - n {\bar{x}} ^2 \right) = \frac{1}{n-1} \left\{ \sum_{i=1}^n x_i^2 - \frac{1}{n}\left(\sum_{i=1}^n x_i\right)^2 \right\}\end{aligned}, \end{equation}\] which initially looks more complicated but is actually simpler calculate: compute first the sums \(\sum_i x_i\) and \(\sum_i x_i^2\) then the rest is easy.
The alternative form is easy to derive as shown in Video 3. However, before you watch the video, try to derive the result yourself.
Video 3: Derivation of the variance
Standard deviation. The standard deviation, typically denoted \(s\), is just the positive square root of the variance \(s^2\). It is therefore the root-mean-square deviation of the observations about their sample mean. The standard deviation is in the same units as the original measurements and for this reason it is preferred to the variance as a descriptive measure. However, it is often easier from a theoretical and computational point of view to work with variances. Thus the two measures are complementary.
Range. This is simply the difference between the largest and smallest observation. It can be very useful for comparing the variability in samples of equal size but it is unfortunately affected by the number of observations; the range usually increases as the number of observations in the data set increases.
Inter-quartile range (IQR). The quartiles are chosen so that 25% of the data lies below the lower quartile (\(Q_1\)) and 25% above the upper quartile (\(Q_3\)). Thus the lower quartile, median and upper quartile split the data into four equal portions.
Min 25% \(Q_1\) 25% \(Q_2\) 25% \(Q_3\) 25% Max After the data are arranged in ascending order, the lower quartile is calculated as the \(\frac{(n+1)}{4}\)th observation and the upper quartile is calculated as the \(\frac{3(n+1)}{4}\)th observation. When \(n + 1\) is not divisible by 4 then the quartiles are calculated by performing some simple interpolation rule (see Example 1). The difference between the lower and upper quartiles is called the interquartile range and it measures the range of the central 50% of the sample.
Quartiles and IQR for Cavendish’s data.
Here \(n = 29\), so \(Q_1\) is the ‘\(7.5\)th’ observation. The \(7\)th observation is \(5.29\) and the \(8\)th observation is \(5.30\), so \(Q_1 = 5.295\).
Similarly, \(Q_3\) is the ‘\(22.5\)th’ observation. The \(22\)nd observation is \(5.61\) and the \(23\)rd observation \(5.62\), so \(Q_3 = 5.615\).
The IQR is \(5.615 - 5.295 = 0.320\).
Link to Cavendish dataset.
2.3 Robustness of summary statistics
In many circumstances (for example, when the distribution of observations is unimodal and roughly symmetric/unskewed and no outliers exist–see definitions in Section 3–visualisation of data) then the mean, median and mode will be very close together. However, if for example the distribution is very skewed there may be a considerable differences between the three measures. The median is often considered as a better description of location for asymmetric distributions than the mean and is much more robust to outliers. In other words, the value of the mean is sensitive to the values of very large or very small observations, whereas the median is not. An amended version of the mean, define to be more robust, is the trimmed mean.
The (\(\alpha\)%)-trimmed mean is computed by deleting the \(\alpha\)% smallest observations and the \(\alpha\)% largest observations before calculating the mean of what is remaining. Trimmed means are used in many judge based scoring systems such as diving and figure skating.
This is more robust than the mean, but the choice of \(\alpha\) is somewhat arbitrary and for this reason the median is usually preferred.
Robustness.
Let’s calculate the mean, median and 10%-trimmed mean for the following data:
\[14, 19, 13, 10, 18, 20, 15, 16, 11, 100\]
The mean = \(23.6\), and median = \((15+16)/2 = 15.5\). Notice how the value 100 seems to be an outlier and that this has affected the mean, which is no longer very representative of the “location” of the data.
For the trimmed mean, we delete the bottom and top 10% of observations and compute the mean of what’s left, i.e., we delete 10 and 100, then we get that:So by “trimming” we’ve gotten rid of the influence of the outlier 100, and the trimmed mean seems a better measure of location compared with the mean.
The median is robust to outliers, whereas the mean is not. In Video 4 we demonstrate this using the Cavendish dataset. There is also a link provided to the R Shiny app used in the video so that you can explore the data and trimming for yourself.
Video 4: Robust measures
We talk about robustness not only for measures of location but also, for example, for measures of spread.
The interquartile range is very robust to outliers, whereas the standard deviation is very sensitive to outliers.