BBS1003 Statistics

BBS1003 Statistics - All information summarised - Maastricht University

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SUMMARY BBS1003

Week 1
Statistics works on the basis of creating a statistical model to predict what will happen.
The type of variable determines which statistical technique can be used in a proper way.
 Variable: recorded piece of information or a characteristic about a person in a study
- Specific for each subject or individual, the opposite is a constant
o Binary variable: variables which only take two values (true or false, yes or no)
 Identifiers: identify an individual (student number, employer ID) but are not variables
 Categorical (qualitative) variable: people or things are placed in groups or categories
- Number of categories may be determined naturally or chosen by the researcher
 Nominal, Ordinal
 Numeric (quantitive) variables: numbers
o Discrete: inferiorly, describes a whole number
- e.g., number of births per day
o Continuous: measured on a continuous scale (with decimals)
- e.g., age, weight, income, temperature, time
 Interval, Ratio
Numeric variables can always be converted into categorical variables Levels of Data
1. Nominal: ordering is not based on magnitude or size (e.g., disease, sex, country of birth, colour)
- There is no ranking order and the space between the scores does not have any meaning
- Presented in bar chart
- The only average that can be taken is Mode
2. Ordinal: ordering based on magnitude or size, there is a ranking order (e.g., size of coffee, (dis)agree)
- The space between the rankings is not always the same
3. Ratio-scale: 0 is meaningful = 0 means absence
4. Interval-scale: 0 has no meaning = 0 does not mean absence
- Ratio and Interval data are summarized in a line graph.
Interval variables contain the same information as nominal and ordinal variables plus the extra information
that differences between scores can be meaningfully interpreted.

Summarizing Data
 Frequency Table: percentages show distribution
- For a small group, frequency is more meaningful
- For a bigger group, proportion and percentages are more
meaningful
- Most of the times, there is looked at the cumulative percentage,
which sums all percentages.

 Bar chart: separate bars to indicate separate categories
with no continuity between them
 Qualitative, Nominal/ Ordinal Variables
- The frequency (counts) at the Y-axis and the score at the X-axis
- The bars are not connected and the distance between the bars does not have a meaning  Pie chart: total is represented by whole circle]
 Histogram: non-separate bars dealing with continuous data
- Before drawing a histogram, look at a table showing class intervals, boundaries, widths and
frequencies  Do not look at height of the bar, but the surface area of the bar.
- The width of each bar is meaningful  Chosen by the researcher itself
- The height of each bar is calculated as frequency / class width
- The total area under all bars will be exactly equal to one
 Quantitive, Interval and Ratio Variables

The left boundary of each class is included in the class interval ([) and the right boundary is an open
boundary meaning that the boundary is not included in the class interval = (.
Measures of Central Tendency
 Average: a ‘typical’ or ‘middle’ number
 Central tendency: central or typical value for a probability
distribution.
- Represent more numbers within one number (4 3 1 6 1 7) o Mean: sum of all numbers divided by total of numbers
- Used for only quantitive data (numbers) which is fairly symmetrical
- Can be seen as the centre of mass/gravity of a massive piece of
wood in the shape of the distribution and the area under the curve.
o Median: order all the numbers from low to high, find the middle
number
- Used for only quantitive data, when the data is skewed.
 Even total of numbers: take middle number
 Uneven: two middle numbers, take the mean of these. o Mode: most common number in a data set, most frequent value
 When there is no most common number, there is no mode.

Measures of Spread
The variance and Standard Deviation are
single measures which summarize
differences between scores.
 It should be noted that both of these
summary measures represent exactly the
same information.  Variance: measure of how peeked/flat the
distribution is
- Average of the squared differences from the
mean (Standard Deviations)
- Where the variance gives a rough idea of the
spread, the Standard Deviation is more
concrete
 Standard Deviation: just the square root of the
variance.

Changes to Variance and Mean
- If all values go through a change (e.g all increase by 19 or multiplied by 2), the Mean also increases by
this exact amount or factor.
- The Variance does not change by the addition or subtraction to values.
 If values are multiplied by X, the variance will increase by X^2, as the values are squared in the
variance equation. There is a difference in calculating the variance when you have a sample instead of a whole population.
In these cases, divide the variance by N-1.
This takes into account how different the sample may be from the whole population. Normal Distribution
 Normal Distribution: how large data sets will look when plotted
- Mean = Median = Mode  At the centre of the normal distribution
- Unimodal: only 1 peak in distribution
- Symmetrical when divided into two
 Standard Deviation: average distance between one observation and the Mean
- Spread of the normal distribution, can be either above or below the Mean
- Used to measure how much variation exists in distribution
 Low standard deviation: mean values are close to the mean = lower variability
 High standard deviation: mean values spread out over a large range = higher variability
Σ (𝑋 − 𝑀)2
𝑆𝐷 = √
𝑛 − 1
For the Standard Deviation, there is the 68/95/99.7 – 1/2/3 SD rule.
- If we extend 1 SD above the mean (+) and 1 SD below the mean (-), approximately 68% of the
observations are contained within this interval.
- Approximately 95% of the population would be between 2 SD (Standard Deviations) above the mean
and 2 SD below the mean for a normal distribution How many Standard Deviations an observed value is from the Mean, is represented in a Z-score.
 Standardize = Unit conversion to SD?
 Z-score: used to measure how many standard deviations above or below the mean a particular
score is
Modality, Skewness and Kurtosis
 Modality: number of peaks in distribution
- Most distributions are normal (unimodal), but they can also be bimodal or multimodal
 Skewness: measure of symmetry of distributions
- When the peak is off-centre, on tale of the distribution is larger than the other
- Every time, the total area under the curve will be exactly equal to 1. 𝑀𝑒𝑎𝑛 − 𝑀𝑒𝑑𝑖𝑎𝑛
𝑆𝑤𝑒𝑘𝑛𝑒𝑠𝑠 =
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
o Positive skewness: graph goes to the left
- Leaving tail pointing to right  Positive direction
- Mode < Median < Mean o Negative skewness: graph goes to the right
- Leaving tail pointing to the left
 Negative direction
- Mean < Median > Mode - Positive skewness is more common than negative skewness.
- Skewness affects the Central Tendency and therefore the mode/median/mean are not in the same place.
- Normal distributions are symmetrical and well-shaped, with skewness = 0, because the mean equals de
median.  Kurtosis: measuring shape of the curve
o Mesokurtic: normal distribution, K = 0
o Leptokurtic (positive kurtosis): peaks sharpy with flat tails
- Less variability, K > 0
o Platykurtic (negative kurtosis): flattened graph, highly
dispersed, K< 0
 Boxplot: shows median, IQR and range all in one diagram
o Median (Q2): point that cuts dataset in half
- In the box, there is 50% of all observations (75% - 25%)