Download Bell-Shaped Curve (B

Bell-Shaped Curve.
The bell-shaped curve is the term used to describe the shape of a normal
distribution when it is plotted with the x axis showing the different values in
the distribution and the y axis showing the frequency of their occurrence. The
bell-shaped curve is a symmetric distribution such that the highest frequencies
cluster around the mid-point of the distribution with a gradual tailing off
towards 0 at an equal rate on either side in the frequency of values as they
move away from the centre of the distribution. In effect, it resembles a church
bell, hence the name. Figure 1 illustrates such a bell-shaped curve. As you
can see from the symmetry and shape of the curve all three measures of central
tendency —the mean, mode and the median—coincide at the highest point of
the curve.
bottom .025
of cases
top .025
of cases
mean
μ-1.96*σ
μ+1.96*σ
The bell shaped curve is described by its mean, μ, and its standard deviation,
σ. Each bell-shaped curve with a particular μ and σ will represent a unique
distribution. As the frequency of distributions is greater towards the middle of
the curve and around the mean, the probability that any single observation
from a bell-shaped distribution will fall near to the mean is much greater than
that it will fall in one of the tails. As a result, we know, from normal
probabilities, that in the bell-shaped curve 68 per cent of values will fall within
one standard deviation of the mean, 95 per cent will fall within roughly two
standard deviations of the mean and nearly all will fall within three standard
deviations of the mean. The remaining observations will be shared between
the two tails of the distribution. This is illustrated in Figure 1, where we can
see that 2.5 per cent of cases fall into the tails beyond the range represented by
1
μ ± 1.96*σ. This gives us the probability that in an approximately bell-shaped
sampling distribution any case will fall with 95 per cent probability within this
range, and thus by statistical extrapolation the population parameter can be
predicted as falling within such a range with 95 per cent confidence.
A particular form of the bell-shaped curve has its mean at 0 and a standard
deviation of 1. This is known as the standard normal distribution, and for such
a distribution the distribution probabilities represented by the equation μ ± z*σ
simplify to the value of the multiple of σ (or Z-SCORE) itself. Thus, for a
standard normal distribution the 95 per cent probabilities lie within the
range ±1.96.
Bell-shaped distributions, then, clearly have particular qualities deriving from
their symmetry, such that it is possible to make statistical inferences for any
distributions which approximate this shape. By extrapolation, they also form
the basis of statistical inference even when distributions are not bell-shaped.
In addition, distributions that are not bell-shaped can often by transformed to
create an approximately bell-shaped curve. Thus, for example, income
distributions, which show a skew to the right, can be transformed into an
approximately symmetrical distribution by taking the log of the values.
Conversely, distributions with a skew to the left, such as examination scores or
life expectancy, can be transformed to approximate a bell-shape by squaring or
cubing the values.
2