# Measures of Dispersion

### INTRODUCTION   TO DISPERSION

In order to get an idea of the general level of values in a frequency distribution, we have studied the various  measures  of  location   that  are  available.  However,   the  figures  which  go  to  make  up  a distribution may all be very close to the central  value, or they may be widely dispersed  about  it, e.g. the mean of 49 and 51 is 50, but the mean of 0 and 100 is also 50!  You can see, therefore, that two distributions may have the same mean but the individual values may be spread  about  the mean  in vastly different  ways.

When applying  statistical  methods  to practical  problems,  a knowledge  of this spread  (which  we call “dispersion” or “variation”) is of great importance.  Examine the figures in the following  table:

Although  the two factories  have the same mean output, they are very different  in their week-to-week consistency.    Factory  A achieves  its mean  production with  only  very little  variation from  week  to week, whereas Factory B achieves the same mean by erratic upsand-downs from week to week.  This example shows that a mean (or other measure of location)  does not, by itself, tell the whole story and we therefore  need to supplement it with a “measure of dispersion”.

As was the case with measures of location,  there are several different measures of dispersion  in use by statisticians. Each  has  its own  particular merits  and  demerits,  which  will be discussed  later. The measures in common  use are:

− Range

− Quartile deviation

− Mean  deviation

− Standard deviation

We will discuss three of these here.

### THE RANGE

This is the  simplest  measure  of dispersion;  it is simply  the  difference  between  the  largest  and  the smallest. In the example  just given, we can see that  the lowest  weekly output for Factory  A was 90 and the highest was 107;  the range  is therefore  17. For Factory  B the range  is 156  – 36 = 120.  The larger range for Factory  B shows that  it performs  less consistently  than  Factory  A.

The advantage of the range  as a measure  of the dispersion  of a distribution is that  it is very easy to calculate and its meaning  is easy to understand. For these reasons  it is used a great deal in industrial quality control work. Its disadvantage is that it is based on only two of the individual  values and takes no  account   of  all  those  in  between.   As a  result,  one  or  two  extreme  results  can  make  it  quite unrepresentative. Consequently, the range is not much used except in the case just mentioned.

#### THE QUARTILE DEVIATION, DECILES AND  PERCENTILES

##### The Quartile Deviation

This measure of dispersion is sometimes called the “semi-interquartile range”. To understand it, you must cast your mind back to the method of obtaining  the median from the ogive. The median, you remember, is the value which divides the total frequency into two halves. The values which divide the total frequency into quarters  are called quartiles and they can also be found from the ogive

##### Calculation of the Quartile Deviation

The quartile  deviation  is not  difficult  to calculate  and  some examination questions  may specifically ask for it to be calculated, in which case a graphical  method  is not acceptable. Graphical methods  are never quite as accurate  as calculations.

##### Deciles and Percentiles

It  is  sometimes  convenient,   particularly when  dealing  with  wages  and  employment   statistics,  to consider values similar  to the quartiles  but  which divide the distribution more  finely. Such partition values are deciles and  percentiles.  From their  names you will probably have guessed that  the deciles are the values which  divide the total  frequency  into  tenths  and  the percentiles  are the values which divide the total  frequency  into hundredths. Obviously  it is only meaningful  to consider  such values when we have a large total  frequency.

#### THE STANDARD   DEVIATION

Most  important of the measures  of dispersion  is the standard deviation.  Except  for the use of the range in statistical  quality control and the use of the quartile  deviation  in wages statistics, the standard deviation  is used  almost  exclusively  in  statistical  practice.  It  is defined  as the  square root  of the variance and so we need to know  how to calculate  the variance  first.

##### The Variance

We  start  by  finding  the  deviations  from  the  mean,  and  then  squaring   them,  which  removes  the negative signs in a mathematically acceptable  fashion,

##### Standard Deviation of a Simple Frequency Distribution

If the data  had  been given as a frequency  distribution (as is often  the case) then  only the different values would  appear  in the “x” column  and we would  have to remember  to multiply  each result by its frequency:

##### Standard Deviation of a Grouped Frequency Distribution

When we come to the problem  of finding the standard deviation  of a grouped  frequency distribution, we again assume that  all the readings  in a given group  fall at the mid-point of the group,  so we can find the arithmetic mean as before

##### Characteristics of the Standard Deviation

In spite of the apparently complicated  method  of calculation, the standard deviation  is the measure of dispersion used in all but  the very simplest  of statistical  studies.  It is based  on all of the individual  items, it gives slightly more emphasis to the larger deviations but does not ignore the smaller ones and, most important, it can be treated  mathematically in more advanced  statistics.

#### THE COEFFICIENT  OF  VARIATION

Suppose that  we are comparing  the profits  earned  by two  businesses.  One  of them  may be a fairly large business  with  average  monthly  profits  of RWF50,000, while the other  may be a small firm with average monthly profits of only RWF2,000. Clearly, the general level of profits is very different in the two cases, but  what  about  the month-by-month variability?  We will compare  the two  firms as to their variability  by calculating the two  standard deviations;  let us suppose  that  they both  come to RWF500. Now,  RWF500 is a much  more significant  amount in relation  to the small firm than  it is in relation  to the large firm so that,  although they have the same standard deviations, it would  be unrealistic  to say that   the  two  businesses  are  equally  consistent   in  their  month-to-month earnings  of  profits.   To overcome  the difficulty,  we express the SD as a percentage  of the mean in each case and we call the result the “coefficient  of variation”.

Applying the idea to the figures which we have just quoted, we get coefficients of variation (usually indicated in formulae  by V or CV) as follows:

This shows that,  relatively speaking,  the small firm is more erratic  in its earnings  than  the large firm.

Note that  although a standard deviation  has the same units as the variate,  the coefficient of variation is a ratio and thus has no units.

Another  application of the coefficient  of variation comes when  we try to compare  distributions the data  of which are in different  units as, for example,  when we try to compare  a French business with an American business.  To avoid  the trouble  of converting  the dollars  to euro  (or vice versa) we can calculate the coefficients of variation in each case and thus obtain  comparable measures of dispersion.

SKEWNESS

When  the  items  in a distribution are  dispersed  equally  on  each  side of the  mean,  we say that  the distribution is symmetrical.  Figure 6.2 shows two symmetrical  distributions.

When the items are not symmetrically dispersed on each side of the mean, we say that the distribution is skew or asymmetric.

A distribution which has a tail drawn  out to the right is said to be positively skew, while one with a tail to the left, is negatively skew. Two distributions may have the same mean and the same standard deviation  but  they  may be differently  skewed.  This will be obvious  if you look  at one of the skew distributions in Figure 6.3 and then look at the same one through from the other  side of the paper!

What,  then,  does skewness  tell us? It tells us that  we are to expect  a few unusually  high values in a positively skew distribution or a few unusually  low values in a negatively skew distribution.

If a distribution is symmetrical,  the mean,  mode and median  all occur at the same point,  i.e. right in the middle. But in a skew distribution the mean and the median  lie somewhere  along the side of the “tail”,  although the  mode  is still  at  the  point  where  the  curve  is highest.  The  more  skewed  the distribution, the greater  the distance  from  the mode  to the mean and  the median,  but  these two  are always  in the same order;  working  outwards from  the mode,  the median  comes first and  then  the mean – see Figure 6.4.

For  most  distributions,  except  for  those  with  very  long  tails,  the  following   relationship  holds approximately:

Mean – Mode  = 3(Mean  – Median)

The more skew the distribution, the more spread  out are these three measures  of location,  and so we can use the amount of this spread  to measure  the amount of skewness. The most usual way of doing this is to calculate

You  are  expected   to  use  one  of  these  formulae   when  an  examiner   asks  for  the  skewness  (or “coefficient  of skewness”, as some of them  call it) of a distribution. When  you do the calculation, remember to get the correct  sign (+ or –) when subtracting the mode  or median  from  the mean and then  you  will  get  negative  answers  from  negatively  skew  distributions, and  positive  answers  for positively skew distributions. The value of the coefficient of skewness is between –3 and +3, although values below –1 and above +1 are rare and indicate  very skewed distributions.

Examples  of variates  with  positive  skew  distributions include  size of incomes  of a large  group  of workers, size of households, length  of service in an organisation, and  age of a workforce. Negative skew distributions occur less frequently.  One such example is the age at death for the adult population in Rwanda.

#### AVERAGES AND MEASURES OF DISPERSION

##### Measures of Central Tendency and Dispersion
• Averages and variations for ungrouped and grouped data.
• Special cases such as the Harmonic mean and the geometric mean

In the last section we described data using graphs, histograms and Ogives mainly for grouped numerical data. Sometimes we do not want a graph; we want one figure to describe the data.

One such figure is called the average. There are three different averages, all summarise the data with just one figure but each one has  a different interpretation.

When describing data the most obvious way and the most common way is to get an average figure. If I said the average amount of alcohol consumed by Rwandan women is 2.6 units per week then how useful is this information? Usually averages on their own are not much use; you also need a measure of how spread out the data is. We will deal with the spread of the data later.

What is the best average, if any, to use in each of the following situations? Justify each of your answers.

• To establish a typical wage to be used by an employer in wage negotiations for a small company of 300 employees, a few of whom are very highly paid specialists.
• To determine the height to construct a bridge (not a draw bridge) where the distribution of the heights of all ships which would pass under is known and is skewed to the right.

There are THREE different measures of AVEARGE, and three different measures of dispersion. Once you know the mean and the standard deviation you can tell much more about the data than if you have the average only.

The Median and the Quartiles.

The median is the figure where half the values of the data set lie below this figure & half above. In a class of students the median age would be the age of the person where half the class is younger than this person and half older. It is the age of the middle aged student.

If you had a class of 11 students, to find the median age, you would line up all the students starting with the youngest to the oldest. You would then count up to the middle person, the 5th one along, ask them their age and that is the median age.

To find the median of raw data you need to firstly rank the figures from smallest to highest and then choose the middle figure.

For grouped data it is not as easy to rank the data because you don’t have single figures you have groups. There is a formula which can be used or the median can be found from the ogive. From the ogive, you go to the half way point on the vertical axis (if this is already in percentages then up to 50%) and then read the median off the horizontal axis.

##### The Mode

There is no measure of dispersion associated with the mode.

The mode is the most frequently occurring figure in a data set. There is often no mode particularly with continuous data or there could be a few modes. For raw data you find the mode by looking at the data as before, or by doing a tally.

For grouped data you can estimate the mode from a histogram by finding the class with the highest frequency and then estimating.

• Measures of dispersion- range, variance, standard deviation, co-efficient of variation.

The range is explained earlier it is found crudely by taking the highest figure in the data set and subtracting the lowest figure.

The variance is very similar to the standard deviation and measures the spread of the data. If I had two different classes and the mean result in both classes was the same, but the variance was higher in class B then results in class B were more spread out. The variance is found by getting the standard deviation and squaring it.

The standard deviation is done already.

The co-efficient of variation is used to establish which of two sets of data is relatively more variable.

For example, take two companies ABC  and CBA. You are given the following information about their share price and the standard deviation of share price over the past year.

The Harmonic mean: The harmonic mean is used in particular circumstances namely when data consists of a set of rates such as prices, speed or productivity.

Dispersion and Skewness:

The normal distribution is used frequently in statistics. It is not skewed and the mean, median and the mode will all have the same value. So for normally distributed data it does not matter which measure of average you use as they are all the same.

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