Statistics

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Bstats Notes


Business Statistics (University of Technology Sydney)




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LECTURE 1: INTRODUCTION AND DESCRIPTIVE
S TAT I S T I C S I

TYPES OF DATA


QUALITATIVE/CATEGORICAL

 Mutually exclusive labels (one label cannot mean two things)
 Not often numbers, if so, numbers have no mathematical meaning
- Nominal: ordering/ranking makes no sense, numerical labels are arbitrary
- Ordinal: ordering/ranking has meaning/can be interpreted, numerical labels respect
the ordering

QUANTITATIVE/NUMERICAL

 Numbers used to record certain events, numbers have mathematical meaning
- Interval: quantity in difference is meaningful, but in ratio is not; zero has no natural
meaning
- Ratio: difference and ratio of two quantities is also meaningful; zero is meaningful

WORKING WITH CATEGORICAL DATA


 Intuitive to tabulate and visualise, technique is frequency distribution
 Frequency counts: total no of occurrences for each category
 Relative frequency: fraction/proportion of the total no of data items belonging to that
category
 Percent frequency: relative frequency x 100 (%)
 Excel function COUNTIF, technique to use is frequency counts
 To visualise: histogram (categories on x-axis, frequency/relative frequency/percent
frequency on y-axis) or pie chart

INTERMEZZO: THE LANGUAGE


 Random variable (r.v.): a variable whose value appears randomly
- usually denoted by capital letters
- Realisations/observations of an r.v. are denoted by lowercase letters
- e.g. N and n denote the size/number of observations - N is referred to population size,
n denotes sample size (no of data points collected in a sample)
 Population: collection of people, objects or items of interest; complete pool of certain
random variable
 Sample: subset of a population; random collection of a certain size from the population
 Probability distribution: general shape of probability for values that a random variable
may assume

DESCRIPTIVE STATISTIC: CENTRAL TENDENCY




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 Measure of central tendency yields info about the centre of a set of numbers (distribution
of a r.v.’s) – does not focus on the span of the dataset or how far values are from middle
numbers
 gives an idea of what a typical, middle, or average that a r.v. can take
 sometimes called measures of location

THREE MEASURES OF CENTRAL TENDENCY
Mode - most frequently occurring value in a set of data
- In case of a tie for the most frequently occurring value, two modes are listed
and the data is said to be bimodal
- Datasets with two or more modes are referred to as multimodal
- Concept of mode is often used in determining sizes
- Appropriate descriptive summary measure for categorical data

Media - middle value in an ordered array of numbers
n n+1
- A way to locate the median is by finding the th term in the ordered array
2
- Large and small values do not inordinately influence the median – hence the
best measure of location to use in the analysis of variables in which extreme but
acceptable values can occur at just one end of the data
- Not all info from the dataset is used
- Data must be quantitative or be able to be ranked
Mean - Average of a set of numbers
- Sample mean is represented by X
- Population mean is represented by 
- Data should be quantitative as it needs to be summed
- Affected by all values – advantage because it reflects all the data, but
disadvantage because extreme values pull the mean towards extremes



 Can consider population mean or sample mean – if you denote r.v. by X , you have:
- Population mean is denoted by  or E( X) , computed by




- Sample mean is denoted by X , computed by





Outlier: observation of the r.v. of interest whose value is far outside the range of other
realisations – often biases impressions about the distribution of r.v. in the dataset, we
may want to correct for such biases/simply remove such a data point




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DESCRIPTIVE STATISTIC: VARIABILITY


 Measures of variability yield info about the likelihood of a realisation of the r.v. is away
from the centre of its distribution, describes the spread/dispersion of a dataset
 Gives an idea of fluctuation and volatility across realisations of the r.v.
 The more variability in a dataset, the less typical they are of the whole set
 Using measures of variability in conjunction with measures of central tendency makes
possible a more complete numerical description of the data (measure of variability is
necessary to complement the mean value when describing data)

FIVE MEASURES OF VARIABILITY
Range - Maximum – minimum
- Crude measure of variability
- Advantage: ease of calculation; disadvantage: affected by extreme
values (thus application as a measure of variability is limited)
Inter-quartile - Distance between the first and third quartiles, IQR = Q 3−Q 1
range - Essentially the range of the middle 50% of the data
- useful when there is interest in values towards the middle rather
than values in the extremes
Variance - one is obtained from the other, they are presented together
- Variance and standard deviation measure out how spread out a r.v.
Standard is, the large the more spread out
deviation - involves considering how far each data value is from the mean and
describing this dispersion on average
- subtracting the mean from each value of data yields the deviation
from the mean: x−¿ - negative deviations represent values below
the mean, positive deviations represent values above the mean




VARIANCE
- Average squared distance between data points and their mean
- Sum of squared deviations from the mean of a set of values is called
the sum of squares of x : SS x

STANDARD DEVIATION
- Square root of the variance – has the same unit of the original data
- Estimate of the average distance that individual values are away
from the mean




Coefficient of - Standard deviation ÷ mean
variation




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