Mathematics & Modelling Flashcards
What are the terms given to units with factors smaller than 1 i.e. 10 to power of -1, -2, -3, -6, -9, -12 and -15?
- 1 deci (d)
- 2 centi (c)
- 3 milli (m)
- 6 micro (µ)
- 9 nano (n)
- 12 pico (p)
- 15 femto (f)
What are the terms given to units with factors larger than 1 i.e. 10 to power of 1, 2, 3, 6, 9, 12 and 15?
1 deca (da) 2 hecto (h) 3 kilo (k) 6 mega (M) 9 giga (G) 12 tera (T) 15 peta (P)
Define population, sample, variable and observation in the context of statistics.
Population - the individuals collected, subject to investigation
Sample - a sub-set of the population that theoretically represent the whole population
Variable - differing characteristics
Observation - a single measurement part of sample; a single unit
What is sampling? Give 4 important things to note, and the 5 types.
Scale (intensive/extensive)
Primary/secondary data
Size
Sources of error
- Stratified
- Systematic
- Volunteer
- Opportunity
- Random
Define discontinous (discrete) data and the associated scales of measurement; give examples.
Involves integers (whole numbers) and counts; where fractions are not possible
Nominal = categorically discrete, no numerical meaning, can be coded
- Weakest, the most ‘elementary’
- E.g. gender, colour
Ordinal = quantities exhibiting a natural ordering (rank) but where arithmetic isn’t possible
- Discrete, categorical; involve classification and labels
- Cannot state that intervals between values are equal
- E.g. Likert scales (1-5)
Define continous data and the associated scales of measurement; give examples.
Values at any point along an uninterrupted scale; more powerful
Interval = where intervals evenly spaced between values but has NO absolute zero
- Ability to add and subject but not multiply or divide
- E.g. temperature, dates
Ratio = natural absolute zero
- All forms of arithmetic possible
- The most powerful form of measurement (highest)
- E.g. percentages, length, mass, time
What ways can discontinous and continuous data be represented/visualised?
Discontinuous (discrete) = bar graphs; categories on x-axis
Continuous = histograms; area of bars proportional to data represented (frequency = fd x cw)
Define the measures of central tendency
Mean - add up, divide by no. of data points
Median - middle
Mode - most common
Range - smallest from largest
What are Measures of Dispersion?
Measuring the spread of data from a central point (mean value) = Standard Deviation
- Gives us the Variance (mean square deviation)
- The larger the SD, the more inconsistent and spread out the data is
How is distribution described besides the SD/variance? Refer to skewness and kurtosis.
Skewness = measure of symmetry
- Symmetrical ‘bell curve’ is called Normal Distribution
- Positive skew; skewed to left = mode < mean < median
- Negative skew; to right = mode > mean > median
Kurtosis = measure of peakedness
- Meso ~3
- Lepto >3
- Platy <3
Outline the 4 moments of a distribution that must be acknowledged in order for an understanding of what can be done with the data.
1st Central value (mean)
2nd Spread around mean (SD/variance)
3rd Skewness
4th Kurtosis
What 3 things determine whether a PARAMETRIC or NON-PARAMETRIC test is used?
- Scale of measurement
- Degree of normality
- Sample size
Outline the assumptions for the two test types
Parametric:
- Interval or ratio data
- Normally distributed
- 30 or more observations
Non-Parametric:
- All nominal or ordinal data
- Interval/ratio that is not normally distributed
- Small sample size (less than 30)
Give the 3 types of probability.
- Subjective
- Theoretical
- Experimental
Define sample space (n) and event (the 2 types)
Sample Space (n) = possible outcomes
Event = subset of sample space
- Mutually exclusive e.g. dice
- Independent e.g. cards
Define the Null and Alternative Hypotheses; on what basis are they rejected/accepted?
Null (H0) = No statistical significance, no change
Alternative (H1) = THere is a statistical significance
If test statistic exceeds the probability threshold at a certain significance level, then we have to accept the null. If it is lower, you can reject it.
100% to chance: 0%, p=1
5% to chance: 95%, p = 0.05
1% to chance: 99%, p = 0.01
0.1% to chance: 99.9%, p = 0.001
What is the Kolmogorov-Smirnov Test?
To test the relationship between a normal curve and another curve (observed vs. expected curve)
Null hypothesis = normal distribution
If null rejected, the distribution is not normal, so a non-parametric test must be used.
Parametric statistics: inferential tests (x2)
Tests of difference involving different samples with the same variables and scales.
- t-test (one-sample) = 1 sample compared to a single population for analysis on particular variable
- Independent t-test (two-sample) = 2 different samples compared
- ANOVA (Analysis of Variance) = comparisons between and within 3 or more groups/samples
Parametric statistics: relational tests (x1)
Looking for a correlation between two variables; strong and weak positive (+1) relationship or negative (-1)
= Pearson’s r
Non-parametric statistics: inferential tests (x3)
- Chi-Square Test = differences between sample and population (observed vs. expected)
- Can also be two-way = statistical difference between 2 samples
- Makes use of cross-tabulation (invalid if >20% less than 5) - Mann-Whitney U-Test = comparison of means between 2 samples (t-test equivalent but for non-para)
- Kruskal-Wallis Test = comparison of the means for 3 or more samples
Non-parametric statistics: relational tests (x1)
Spearman’s Rank
What are Explanatory Statistics?
Use of REGRESSION to go a step further than relational statistics; attempting to mathematically calculate a causation (since a correlation alone is not enough)
Often used with parametric relationships (Pearson’s r)
What is regression and what does it give us?
Permits us to make a numerical prediction of one variable by reference to another, giving us explanatory power
What are the independent and dependent variables? What axes do they normally go on if a logical direction of causation is established?
Independent = the variable that causes change (x-axis)
Dependent = the variable that is measured as it changes in response to IV (y-axis)
