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Distributions

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1Distributions Why do we need this?To describe/analyse data in an appropriate way
• To exclude human perception of the data
• Assess differences in samples of populations
– Assess the probability of these populations either
being the same (“null hypothesis”), or being different
(“Alternate hypothesis”)
• As in life, there is no certainty resulting from statistical tests, only chance or probability.
2What is ‘Probability’?How frequently we
expect different
outcomes to occur if we
repeat the experiment
over and over…
3Definitions - variable There are two main types of data or variables:• Qualitative/categorical variables is data that is not given numerically
e.g. gender, place of birth, type of car etc.
• Quantitative data is numerical and are two types:
1. Discrete data – which can only take specific numeric values;
e.g. shoe size, number of brothers, number of cars
2. Continuous data – can take any numerical values;
e.g. height, mass, length, time etc.
4Let look at discreate data and a frequency distribution
5Probability distribution
6What about continuous variables?Time for Lecturer A to respond to emails (in
hours and minutes):
Problem: time to respond is a continuous
variable and not two times will be the same!
A ‘frequency distribution’ will be useless as we would have 50 odd times each with a
frequency of 1.
The solution to this problem is a ‘grouped frequency distribution’
7What about continuous variable Time for Lecturer A to respond to emails (in hours and minutes):
8Continuous variables – grouped frequency distribution
9Examining a DistributionCentral tendency is measured in three ways:
– mean, median and mode.
• Mean - average score of a distribution.
• Median is the centre, or middle score within a distribution.
• Mode is the most frequent score within a distribution.
10Measures of Central Tendency Number of mobile numbers on students’ mobile phones Student # of Numbers
11Calculating mean, median and mode
12—yes
13In a normal distributionMean= median=mode
14The Gaussian (aka ‘normal’) distribution• Distributions for continuous variables are called continuous distributions (probability densities)
• Some probability densities have particular importance in statistics. A very important one is shaped like a bell, and
called the normal distribution
• Many naturally-occurring phenomena can be approximated surprisingly well by this distribution
15The ‘normal distribution’The area under the curve equals 1
• The probability of any exact value
of X is 0
• The area under the curve between
two given points on the X-axis is
the probability that a number
chosen at random will fall between
the two points
Varying the Parameters σ and μ, we obtain
Different Normal Distributions.
16The Mathematical Model
17The Mathematical Model
18Finding ProbabilitiesP(c ≤ X ≤ d ) = ?
Probability is the
area under the
curve!
19Finding Probabilities
20What is the Standard Deviation?Standard Deviation is a measure of variability or scatter around a mean
21What is the Standard Deviation?
22Normal DistributionA symmetric distribution defined on the range -∞to + ∞ whose shape is
defined by two parameters, the mean, denoted σ, that centers the
distribution, and the standard deviation, , that determines the spread
of the distribution.
23Normal Distribution
24Standard deviations – spread of data Normal distribution
25Application of the normal distribution – example 1A certain type of storages battery lasts, on average, 3.0 years, with a standard derivation of 0.5 year.
Assuming that the battery lives are normally distributed, find the probability that a given battery will last LESS than 2.3 years
26Application of the normal distribution – example 1
27Standardized normal variableIt would be difficult and tedious to do the calculus every time we had a new set of parameters for µ and σ. So instead, we usually work with the standardized normal distribution, where µ = 0 and σ = 1
The normal random variable of a standard normal
distribution is called a standard score or a z-score. Every
normal random variable X can be transformed into a z score via the following equation:
28Standardized normal variable Every normal random variable X can be transformed into a z score via the following equation:
29Standard Normal Distributionthe standard normal distribution is a special case of the normal
distribution. It is the distribution that occurs when a normal random
variable has a mean of zero and a standard
deviation of one.
30Standard Normal Distribution
31Application of the normal distribution – example 1A certain type of storages battery lasts, on average, 3.0 years, with a standard derivation of 0.5 year. Assuming that the battery lives are normally distributed, find the probability that a given battery will last LESS than 2.3 years
32Application of the normal distribution – example 1
33Application of the normal distribution – example 1
34table of standard probabilities
35Application of the normal distribution – example 3The averages life expectancy in UK is 81
years old with a standard derivation of 10
years. Assuming that life expectancy is
normally distributed, find the probability
that you will live above 90 year old?
36Application of the normal distribution – example 3The average mark for the BIOT1004 module is 75 marks with a standard derivation of 14.2. Assuming that the mark distribution is normally distributed, find the probability that a student will get a mark between 50-60?
37Application of the normal distribution – example 3
38Anytime that you are performing a statistical analysis (t-test, ANOVA and many more), you need tomake sure that you are working
with normally distributed data, or your
analysis will probably not be valid
39back to the example ‘response time’ Is this distribution ‘normal’?
40Skewness negatively skewed
41Skewness normal (no skew)
42Skewness positively skewed
43Looking at symmetry Skewnessthe level of non-symmetry. A
symmetric dataset got a skewness of 0
44Looking at symmetry Kurtosisa measure of the combined
weight of the tails relative to the rest of
the distribution’. Normally distributed data got a value of 0
45Looking at symmetry
46—