Mean

1
Mean

• The most common measure of central tendency is
  the mean which is also referred to as the
  average. The mean is the total of the scores
  divided by the number of scores.
• Lets say our data set is 5 3 54 93 83 22 17 19.

2
Median

• The median is the point that corresponds to the
  score that lies in the middle of the distribution
  when the data are arranged in increasing or
  decreasing numerical order in other words, it
  is the point that divides the distribution in
  half.
• With an odd number of data values, for example
  21, we have
• With an even number of data values, for example
  20, we have

3
Population and Sample
A population is any entire collection of people,
animals, plants or things from which we may
collect data. It is the entire group we are
interested in, which we wish to describe or draw
conclusions about. A sample is a subset from a
larger group (the population). By studying the
sample we hope to draw valid conclusions about
the larger group. The population for a study of
infant health might be all children born in Fiji
in the 1980's. The sample might be all babies
born on 7th May in any of the years.
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Variance

• The population variance gives an idea of how
  widely spread the values of the random variable
  are likely to be the larger the variance, the
  more scattered the observations.
• Variance is symbolised by V(X) or Var(X) or
• The variance of the random variable X is defined
  to be
• E(X) is the expected value of the random variable
  X.

5
Sample Variance

• Sample variance is a measure of the spread of or
  dispersion within a set of sample data. The
  sample variance is
• where is the sample mean
• Lets say our data set is 5 3 54 93 83. The mean
  of this data set is
• The sample variance is

6
Terciles

• Terciles divide data into three categories that
  have the same chance of occurring.
• Example From 1961 to 1990 the Sept to Dec
  rainfall at Entebbe for 10 years were
  below-normal (tercile 1) 200-445 mm, 10 years had
  rainfall near-normal (tercile 2) 445512mm, and
  10 years above-normal (tercile 3) 5121000mm.

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Histogram

• A histogram is a way of summarising data that are
  measured on an interval scale. It divides up the
  range of possible values in a data set into
  classes or groups. For each group, a rectangle is
  constructed with a base length equal to the range
  of values in that specific group, and an area
  proportional to the number of observations
  falling into that group.

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Frequency Distribution
A frequency distribution is a tabular arrangement
of data whereby the data is grouped into
different intervals. Data presented in this
manner are known as grouped data. The relative
frequency distribution is the ratio of the number
of observations in the interval to the total
number of observations. The percentage frequency
distribution is the relative frequencies of each
interval multiplied by 100. The cumulative
frequency distribution is obtained by computing
the cumulative frequency, defined as the total
frequency of all values in the preceding intervals
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Frequency Table

• Example Yield of rice tonnes per hectare
• Frequency Distribution

10
Cumulative Frequency
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Regression Correlation
Correlation is the statistical measure that
quantifies the linear relationship between two
variables. If you look at a scatter plot of two
variables, their correlation is the slope of the
best fitting straight line that can be drawn
through the points Regression is an extension
of correlation analysis that will predict the
value of one variable (the dependent variable)
based on the values of one or more predictor or
independent variables.
y a bx where y is the predicted value
of the dependent variable a is the intercept b
is the slope of the line x is the value of the
independent variable to be predicted
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Correlation

• Correlation is the statistical measure that
  quantifies the linear relationship between two
  variables . The sample correlation coefficient (
  r ) between X and Y is

13
Correlation Cause

• Correlation means that two variables have some
  type of association with each other, such that as
  one variable increases, the other also increases,
  or decreases. But it does not mean that one of
  the variables is the cause of the other.
• Correlations can demonstrate only that a
  relationship exists or does not exist between
  variables, but correlations cannot indicate
  whether or not the relationship is causal.
• Example
• It has been argued that there is a high
  correlation between the increase in juvenile
  delinquency and the increase in the divorce rate
  in recent years. This may be so. This does not,
  however, indicate that the increase in the
  divorce rate has caused the increase in juvenile
  delinquency.

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Correlation Calculations

• Set out a table and calculate S x, S y, S x2, S
  y2, S xy and mean of x and y.

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Calculations of r, b and a
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Coeff. of Determination r2

• The coefficient of determination
• gives the proportion of the fluctuation of one
  variable that is predictable from the other
  variable.
• is the ratio of the explained variation to the
  total variation.
• ranges from 0 lt  r 2 lt 1,  and denotes the
  strength of the linear association between x and
  y. 
• If r 0.922, then r 2 0.850, 85 of the total
  variation in y can be explained by the linear
  relationship between x and y.

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Hypothesis Testing

• A statistical hypothesis is the speculation
  translated into a statement concerning the
  distribution of a defined population.
• The statistical hypothesis under test is often
  referred to as null hypothesis.
• H0 Boys and girls are of equal height
• The alternative hypothesis is a statement of what
  a statistical hypothesis test is set up to
  establish
• H1 Boys are taller than girls

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Type of Error

• In testing a null hypothesis the level of
  significance is the probability of rejecting a
  true hypothesis. Four possible situations are
• The hypothesis is true and it is accepted.
• The hypothesis is true and it is rejected.
• The hypothesis is false and it is accepted.
• The hypothesis is false and it is rejected.

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Type of Error

• A type I error is often considered to be more
  serious, and therefore more important to avoid,
  than a type II error. The probability of a type I
  error can be precisely computed as
• P(type I error) significance
  level
• A type II error is frequently due to sample sizes
  being too small. The probability of a type II
  error is generally unknown, but is symbolised by
  and written
• P(type II error)

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Probability

• A probability provides a quantitative description
  of the likely occurrence of a particular event.
  Probability is conventionally expressed on a
  scale from 0 to 1 a rare event has a probability
  close to 0, a very common event has a probability
  close to 1.
• The probability of drawing a spade from a pack of
  52 well-shuffled playing cards is 13/52 1/4
  0.25
• When tossing a coin, we assume that the results
  'heads' or 'tails' each have equal probabilities
  of 0.5.

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Subjective Probability

• A subjective probability describes an
  individual's personal judgement about how likely
  a particular event is to occur. It is not based
  on any precise computation but is often a
  reasonable assessment by a knowledgeable person.
• Like all probabilities, a subjective probability
  is conventionally expressed on a scale from 0 to
  1 a rare event has a subjective probability
  close to 0, a very common event has a subjective
  probability close to 1.
• A person's subjective probability of an event
  describes his/her degree of belief in the event.
• Example A political commentator suggests that
  the Green Party may win the next election as they
  have put environment on the top of their agenda.

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Independent Events

• Two events are independent if the occurrence of
  one of the events gives us no information about
  whether or not the other event will occur that
  is, the events have no influence on each other.
• In probability theory we say that two events, A
  and B, are independent if the probability that
  they both occur is equal to the product of the
  probabilities of the two individual events, i.e.
• A and B are independent A and C are independent
  and B and C are independent (pair wise
  independence)

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Example of Independent Events

• Suppose that a man and a woman each have a pack
  of 52 playing cards. Each draws a card from
  his/her pack. Find the probability that they each
  draw the ace of clubs.
• We define the events
• A probability that man draws ace of clubs
  1/52
• B probability that woman draws ace of clubs
  1/52
• Clearly events A and B are independent so
• 
  1/52 . 1/52 0.00037
• That is, there is a very small chance that the
  man and the woman will both draw the ace of
  clubs.

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Mutually Exclusive Events

• Two events are mutually exclusive (or disjoint)
  if it is impossible for them to occur together.
• Formally, two events A and B are mutually
  exclusive if and only if
• Examples
• Experiment Rolling a die once
• Sample space S 1,2,3,4,5,6
• Events A 'observe an odd number' 1,3,5
• B 'observe an even number' 2,4,6
• the empty set, so A and
  B are mutually exclusive.
• A subject in a study cannot be both male and
  female, nor can they be aged 20 and 30. A subject
  could however be both male and 20, or both female
  and 30.

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Time Series

• A time series is a sequence of observations that
  are ordered in time (or space). If observations
  are made on some phenomenon throughout time, it
  is most sensible to display the data in the order
  in which they arose, particularly since
  successive observations will probably be
  dependent.
• Time series are best displayed in a scatter plot.
  The series value X is plotted on the vertical
  axis and time t on the horizontal axis. There are
  two kinds of time series data
• Continuous where we have an observation at every
  instant of time, e.g. electrocardiograms. We
  denote this using observation X at time t, X(t).
• Discrete where we have an observation at (usually
  regularly) spaced intervals. We denote this as
  Xt.

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Time Series Plot
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Terms in Time Series
Trend Component Trend is a long term movement in
a time series. A trend pattern exists when there
is a long-term secular increase or decrease in
the data. It is the underlying direction (an
upward or downward tendency) and rate of change
in a time series. The existence of a trend
(linear or non-linear) in the data means that
successive values will be positively correlated
with each other. Cyclical Component A cyclical
pattern exists when the data are influenced by
longer-term fluctuations such as those associated
with the business cycle.
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Terms in Time Series
Seasonal Component Seasonality is defined as a
pattern that repeats itself over fixed intervals
of time. For example, the costs of various types
of fruits and vegetables, unemployment figures
and average daily rainfall, all show marked
seasonal variation. Irregular Component The
irregular component is that left over when the
other components of the series (trend, seasonal
and cyclical) have been accounted for.
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Terms in Time Series

• Smoothing Smoothing techniques are used to
  reduce irregularities (random fluctuations) in
  time series data. They provide a clearer view of
  the true underlying behaviour of the series.
• Exponential Smoothing
• Exponential smoothing is a smoothing technique
  used to reduce irregularities (random
  fluctuations) in time series data, thus providing
  a clearer view of the true underlying behaviour
  of the series.
• Moving average is a form of average that has been
  adjusted to allow for seasonal or cyclical
  components of a time series. Moving average
  smoothing is a smoothing technique used to make
  the long term trends of a time series clearer.

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Terms in Time Series

• Running medians smoothing is a smoothing
  technique analogous to that used for moving
  averages. The purpose of the technique is the
  same, to make a trend clearer by reducing the
  effects of other fluctuations.
• Differencing is a popular and effective method of
  removing trend from a time series. This provides
  a clearer view of the true underlying behaviour
  of the series.
• Autocorrelation is the correlation (relationship)
  between members of a time series of observations,
  such as weekly share prices or interest rates,
  and the same values at a fixed time interval
  later.

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Probability Distribution

• The probability distribution of a discrete random
  variable is a list of probabilities associated
  with each of its possible values. It is also
  sometimes called the probability function or the
  probability mass function.
• More formally, the probability distribution of a
  discrete random variable X is a function which
  gives the probability p(xi) that the random
  variable equals xi, for each value xi
• p(xi) P(Xxi)
• It satisfies the following conditions