Summary
Summary IB149_Week 4_Distributions
- Module
- (IB149)
- Institution
- The University Of Warwick (UoW)
Notes summarising all the content from lectures and includes worked examples to better understand concepts
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DISTRIBUTIONSDISTRIBUTIONAL SHAPES
What is Distribution of Data
● When working with data (e.g. A children’s clothing company has collected a random sample of
heights (cm) of 12 year old girls) we often ask ourselves these three key questions
○ What is the ‘typical’ or ‘average’ height - calculate the mean or median height which
describes the location of the data
○ How ‘spread out’ are the heights? - calculate the SD, range, or IQ this describes the
variability (dispersion) of the data
○ What is the ‘shape’ of the data - how many peaks or modes, is the shape symmetric or
asymmetric, is it positively or negatively skewed?
● Answers these key questions about the location, spread and shape allows us to describe the
distribution of the data
Frequency Distributions vs Probability Distribution
● Frequency Distribution
● When we have a sample of data, we are dealing with a frequency distribution
● We can use the properties of the frequency distribution to make inferences about the
population
○ Using the information from our sample in order to extend it to our wider
population
● A probability distribution is the theoretical counterpart to the frequency distribution
○ It is the distribution of values that we would expect to see if we obtained data from the
entire population
○ Rarely possible to take data from the entire population, we always deal with samples of
datas, we relate information from our sample to the entire population
Continuous probability distributions
● Probability density function
○ The probability distribution of a
continuous random variable is
known as the probability density
function
○ E.g. Continuous probability
distribution
■ Density - the area under
the curve is the total
probability which totals to 1
NORMAL DISTRIBUTION
, Recognise a Normal Distribution
● The normal distribution is a specific theoretical distribution, also known as the ‘Gaussian
distribution’ or the ‘bell curve’
● Features of a normal distribution
○ Symmetrical
○ Unimodal
○ Mean (µ) = Median = Mode
○ Tails asymptotic to the x-axis
● 95% of the data will lie between 2 standard deviations of the
mean and 5% of the values in the tails of the distribution (2.5%
in each tail)
Parameters of Normal Distribution
● Impact of changing the standard deviation on the spread
○ When standard deviation decreases, the spread (less variability) of data is smaller and
therefore a narrower and taller distribution
○ When standard deviation increases, there is a wider spread in data, leading to a flatter and
shorter distribution
● Changing the mean
○ When the mean increases the shape of the distribution remains the same but the whole
distribution shifts right
○ When the mean decreases the whole distribution shifts left
● We can conclude that the shape of the normal distribution is determined by the mean and
standard deviation
○ These are called parameters of the distribution X ~ N (µ, ∂ 2)
○ Every normal distribution curve has a different mean and standard deviation
Standard Normal Distribution
● A standard normal distribution has a mean = 0 and standard deviation = 1, Z ~ N (0,1)
● Every point on any normal distribution curve can be mapped to a point on the standard normal
distribution just by shifting the mean and standard deviation
● Z-scores are the values on the horizontal axis of the standard normal distribution
○ They can be positive or negative
○ Z-scores tell us how many ‘standard deviations’ away from the mean a point is
Z-Scores
● To convert any value of x to its corresponding z-score, we need to calculate
the following: Z = (x-µ)/∂ where µ is the mean and ∂ is the standard
deviation
● 90% of the data is between -1.64 and 1.64 and 95% of the data lies between -1.96 and 1.96,
WORKING WITH Z-VALUES: READING THE Z-TABLE
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