CHAPTER 11 ANALYSING DATA Flashcards
what is the arithmetic mean
The arithmetic mean, also known as the ‘average’, is calculated by dividing the sum of the values in question by the number of values.
what is the median
The median is defined as the middle of a set of values, when arranged in ascending (or descending) order.
what is the mode
The mode or modal value of a data set is that value that occurs most often.
why are measures of spread necessary
it is natural to question the extent to which the single value is representative of the whole set. Through a simple example we shall see that part of the answer to this lies in how ‘spread out’ the individual values are around the average. In particular, we shall study the following measures of spread:
the standard deviation and variance
the coefficient of variation.
what is the standard deviation and its formula
The standard deviation (σ) is a way of measuring how far away on average the data points are from the mean. In other words, they measure average variability about the mean. As such standard deviation is often used with the mean when describing a data set.
1Look at the difference between each data value and the mean
2To get rid of the problem of negative differences cancelling out positive ones, square the results
3Work out the average squared difference (this gives the variance)
4Square root to get the standard deviation
what is the coefficient of variation
The coefficient of variation is a statistical measure of the dispersion of data points in a data series around the mean.
It is calculated as follows:
coefficient of variation = standard deviation/mean
is useful when comparing the degree of variation from one data series to another, even if the means are quite different from each other.
probabilities definition
A probability expresses the likelihood of an event occurring.
If an event is certain to occur, then it has a probability of one.
If an event is impossible, then it has a probability of zero.
how are expected values used in judging the financial outcomes various options
An expected value is a long run average. It is the weighted average of a probability distribution.
While techniques such as EV can help assess the financial aspect of a decision, there are many other non-financial considerations which must be taken account of in any business decision.
Expected value is calculated as follows:
EV = ∑PX
Where X is the outcome and P is the probability of the outcome.
what are the limitations of expected values
A limitation of expected value, which is shared with most other attempts to model reality, is that the outcomes and probabilities need to be estimated. The subsequent analysis can never be more reliable than the estimations upon which it is based.
There is also often a considerable degree of simplification with very limited discrete probability distributions being used. In some cases only subjective estimates of probabilities may be available, and their reliability may be open to question.
In many cases, companies use this technique in one-off decisions. The result in these cases is of little or no use as the activity will only be carried out once and not repeated many times.
Finally, expected values take no account of the decision-makers’ attitude to risk. Avoiding significant downside exposure may be more important than possible gains, although expected values consider each equally.
what is normal distribution and a bell curve
In this case the data is symmetrical and peaks in the centre. This is called a normal distribution. We can draw a line around this distribution to show the shape more clearly. This is called a bell curve.
when are cases of normal distribution found
Normal distributions can be found when we measure things such as:
Exam results
Staff performance gradings
The heights of a group of people etc.
what are the characteristics of a normal distribution
A normal distribution has the following characteristics:
the mean (μ) is shown in the centre of the diagram
the curve is symmetrical about the mean. This means that 50% of the values will be below the mean and 50% of the values will be above the mean.
the mean, median and mode will all be the same for a normal distribution.
The total area under the curve is equal to 1.
To be able to use the normal distribution the distribution must be:
Continuous
Symmetrical
Shaped as a bell curve.
when can normal distribution be used
If we know the mean and the standard deviation for a distribution we can work out the percentage chance (probability) of a certain value occurring.
For example a light bulb manufacturer may want to know how many bulbs will fail after a certain amount of time, or a chocolate bar manufacturer may want to know how many chocolate bars will weigh less than the minimum weight shown on the packaging.
for example if the chocolate bar manufacturer found that 0.05% of bars were lower that the acceptable weight, then 0.05% bars will also be higher than the acceptable weight.
The percentage figures can be obtained using normal distribution tables, which are given in your exam. Note: The tables only show the positive values.
how can you use normal distribution tables in your exam
To use the tables we must first convert our normal distribution to a standard normal distribution.
A standard normal distribution has:
a mean of 0
a standard deviation of 1.
This special distribution is denoted by z and can be calculated as:
z = (x-μ)/σ
Where:
z is the z score
x is the value being considered
μ is the mean
σ is the standard deviation
This calculation is used to convert any value to a standard normal distribution.
