3 Skewness, gaps and multiple peaks

When exploring a data set, four properties worth looking for are outliers, skewness, gaps and multiple peaks. Outliers have been dealt with in some detail above so the comments given below briefly address skewness, gaps and multiple peaks.

3.1 Skewness

If a skewed distribution is represented purely by two numbers, say the mean and standard deviation, then the representation will be inadequate. Remember that the term ‘skewness’ refers to the location of the ‘tail’ of a distribution.

As an example, the data set below gives the current required to burn out a component under test.

The data were obtained by measuring the current in mA applied to an electronic component under conditions of destructive testing, gives the following values for the mean, standard deviation, median and mid-spread:

$\bar{x}=40.72$ $s_{n-1}=11.49$ median $=45.05$ and mid-spread $=9.55$

The values of $\bar{x}$ and $s_{n-1}$ indicate that a lower average current with a greater spread will result in the destruction of the component than that indicated by the median and mid-spread. Clearly, further investigation is necessary to resolve this situation.

3.2 Gaps and multiple peaks

Distributions with gaps and multiple peaks can be very difficult to summarise easily. The stem-and-leaf and box-and-whisker plots shown below summarise some 1972 data concerning adult literacy. The leaves are single digit and the range of achievement reached in the field of literacy ranges from 2% to 100%.

Figure 15

The virtual lack of data between 50 and 60 indicates a gap and suggests that we are in fact dealing with two separate distributions which have the following properties:

1. 2% - 50% literacy having right skew
2. 60% - 100% literacy having left skew.

Notice that the term skewness refers to the tail of a distribution.

The usual summary statistics that you might be tempted to calculate are:

$\bar{x}=54$ and $s_{n-1}=34$ or median $=60$ and mid-spread $=65$

In this case, neither set of statistics is of much use since neither set indicates the gap or the skewness. Without visual representation, a single peaked distribution tends to be assumed, this is, of course, opposite to the truth in this case.

The stem-and-leaf plot is more informative than the box-and-whisker plot since it shows the gap.

In practice we would work with the two constituent distributions and attempt to relate the results in a practical way.

3.3 Final comments on data representations

1. You should not rely on summary statistics such as the mean and standard deviation or median and mid-spread alone to represent a data set. Remember that if a distribution has outliers, gaps, skewness or multiple peaks, then shape is probably more important than location and spread.
2. The shape of a distribution is better shown visually than numerically. Remember that a stem-and-leaf diagram retains the data and arranges the data in rank order and that a box-and-whisker plot emphasises the detail contained in the tails of a distribution.

Exercises

1. The following data give the lifetimes in hours of 50 electric lamps.

   1337	1437	1214	1300	1124	1065	1470	1488	1103	978
   1177	1289	1045	947	969	1339	1594	812	1277	1032
   1167	974	1131	974	1727	1378	1385	1330	1672	1604
   1493	1521	1235	1682	1136	1229	803	1166	1494	1733
   978	1110	1055	1438	1436	1424	766	1283	829	1652

   1. Represent the data using a stem-and-leaf diagram with two-digit leaves.
   2. Calculate the mean lifetime from these data.
   3. Does the mean lifetime give a good indication of the expected lifetime of a lamp?
2. During the winter of 1893/94 Lord Rayleigh conducted an investigation into the density of nitrogen gas taken from various sources. He had previously found discrepancies between the density of nitrogen obtained by chemical decomposition and nitrogen obtained by removing oxygen from air. Lord Rayleigh’s investigations led to the discovery of argon. The raw data obtained during his investigations are given below.

   Date	Source	Weight		Date	Source	Weight
   29/11/93	NO	2.30143		26/12/93	N 2 O	2.29889
   05/12/93	NO	2.29816		28/12/93	N 2 O	2.29940
   06/12/93	NO	2.30182		09/01/94	NH 4 NO 2	2.29849
   08/12/93	NO	2.29890		13/01/94	NH 4 NO 2	2.29889
   12/12/93	Air	2.31017		29/01/94	Air	2.31024
   14/12/93	Air	2.30986		30/01/94	Air	2.31030
   19/12/93	Air	2.31010		01/02/94	Air	2.31028
   22/12/93	Air	2.31001

   1. Organise the data into a frequency table using the classes 2.29-2.30, 2.30-2.31, 2.31-2.32. Draw the histogram representing the data and comment on any unusual features that you may see.
   2. Classify the data according to the two sources ‘Air’ and ‘Other’ . Order each data set and hence find the median, the hinges and the mid-spreads for each data set. Plot box-and-whisker diagrams for the data on a diagram similar to the one shown below.

   

   Comment on any unusual features that you see. What do the box-and-whisker plots tell you about the nitrogen obtained from the two sources?

3. Answer the following questions:
   1. Is the variance measured in the same units as the mean?
   2. Is the mean measured in the same units as the median?
   3. Is the standard deviation measured in the same units as the mode?
   4. Is the mode measured in the same units as the mid-spread?
   5. Is the high-spread measured in the same units as the low-spread?
   6. Is the mid-spread measured in the same units as the hinges?

Answer