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**CS1512 Foundations of Computing Science 2 Lecture 20 Probability and statistics (2)**

© J R W Hunter, 2006

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**Ordinal data a1 ≤ a2 ≤ a3 ≤ ... ≤ ak ≤ ... ≤ aK**

X is an ordinal variable with values: a1, a2, a3, ... ak, aK ‘ordinal’ means that: a1 ≤ a2 ≤ a3 ≤ ≤ ak ≤ ≤ aK cumulative frequency at level k: ck = sum of frequencies of values less than or equal to ak ck = f1 + f2 + f fk = (f1 + f2 + f fk-1 ) + fk = ck fk also (%) cumulative relative frequency

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**CAS marks % relative frequencies % cumulative relative frequencies**

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**Enzyme concentrations**

Concentration Freq Rel.Freq % Cum. Rel. Freq. ≤ c < % ≤ c < % ≤ c < % ≤ c < % ≤ c < % ≤ c < % ≤ c < % ≤ c < % ≤ c < % ≤ c < % Totals

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Cumulative histogram

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**Discrete two variable data**

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**Continuous two variable data**

X Y

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**Time Series Time and space are fundamental (especially time)**

Time series: variation of a particular variable with time

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**Summarising data by numerical means**

Further summarisation (beyond frequencies) Measures of location (Where is the middle?) Mean Median Mode

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**Mean = n _ sum of observed values of X Sample Mean (X) =**

number of observed values x = n use only for quantitative data

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**Sigma xi = x1 + x2 + ... + xi + ... + xn-1 + xn**

Sum of n observations n xi = x1 + x xi xn xn i = 1 If it is clear that the sum is from 1 to n then: x = x1 + x xi xn xn Sum of squares x2 = x x xi xn xn2

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x from frequencies If X is a categorical variable with values: a1, a2, a3, ... ak, aK x = x1 + x xi xn xn (order of summation isn’t important) (e.g. piglets: ) Group together those x’s which have value a1, those with value a2, ... x = x x x x’s which have value a1 - there are f1 of them x x x’s which have value a2 - there are f2 of them x x x’s which have value aK - there are fK of them = f1 * a f2 * a fk * ak fK * aK K = fk * ak k = 1

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Mean Litter size Frequency Cum. Freq ak fk Total K x = fk * ak k = 1 = 1* * *7 + 3* * *10 + 8* *12 + 3*13 + 2*14 = 375 _ X = 375 / 36 = 10.42

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**Median Sample median of X = middle value when n sample observations**

are ranked in increasing order = the ((n + 1)/2)th value n odd: values: , 163, 152, 157 and 157 rank order: 152, 157, 157, 163, 183 median: n even: values: , 173, 180, 164 rank order: 164, 165, 173, 180 median: ( )/2 = 169

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**Median Median = 10.5 Litter size Frequency Cum. Freq 5 1 1 6 0 1 7 2 3**

Total Median = 10.5

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**Median from cumulative distribution**

cumulative % frequency polygon

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**Mode Sample mode = value with highest frequency (may not be unique)**

Litter size Frequency Cum. Freq Mode = 10

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**Skew left skewed symmetric right skewed**

mean < mode mean mode mean > mode

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**Variance Measure of spread: variance**

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**Variance sample variance = s2**

sample standard deviation = s = √ variance

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**Variance and standard deviation**

K x2 = fk * ak2 k = 1 = 1* * * * * * * * *196 = 4145 x = 375 (x)2 / n = 375*375 / 36 = 3906 s2 = ( ) / (36-1) = 6.83 s = 2.6 Litter size Frequency Cum. Freq ak fk Total

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**Piglets Mean = 10.42 Median = 10.5 Mode = 10 Std. devn. = 2.6**

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Quartiles and Range Lower quartile: value such that 25% of observations are below it (Q1). Median: value such that 50% of observations are below (above) it (Q2). Upper quartile: value such that 25% of observations are above it (Q3). Range: the minimum (m) and maximum (M) observations. Box and Whisker plot: m Q1 Q Q M

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Estimating quartiles

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**Linear Regression y y = mc + c x**

Calculate m and c so that (distance of point from line)2 is minimised y y = mc + c x

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**Time Series - Moving Average**

Time Y point MA * * smoothing function can compute median, max, min, std. devn, etc. in window

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