Data Analysis 2

BIOL10272: Practical Techniques

Dr Axel Barlow
email: axel.barlow@ntu.ac.uk

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Data analysis 2

  • Data visualisation
  • Single categorical or ordinal variable: barcharts
  • Single quantitative variable: histograms
  • Data distributions
  • Mean, median, and mode
  • Sample variability: range and standard deviation

Data visualisation

A tool for data exploration and communication

A tool for data exploration and communication

Single categorical or ordinal variable

Barcharts

  • We can visualise a single categorical or ordinal variable using a barchart.
  • Categories go on the x axis and the counts go on the y axis.

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Barchart example

  • Human mitochondrial DNA; 4 nucleotides (A,C,G,T); 16,568 bp
##                                                                                    V1
## 1  GATCACAGGTCTATCACCCTATTAACCACTCACGGGAGCTCTCCATGCATTTGGTATTTTCGTCTGGGGGGTGTGCACGCGA
## 2  TAGCATTGCGAGACGCTGGAGCCGGAGCACCCTATGTCGCAGTATCTGTCTTTGATTCCTGCCCCATCCCATTATTTATCGC
## 3  ACCTACGTTCAATATTACAGGCGAACATACTTACTAAAGTGTGTTAATTAATTAATGCTTGTAGGACATAATAATAACAATT
## 4  GAATGTCTGCACAGCCGCTTTCCACACAGACATCATAACAAAAAATTTCCACCAAACCCCCCCTCCCCCGCTTCTGGCCACA
## 5  GCACTTAAACACATCTCTGCCAAACCCCAAAAACAAAGAACCCTAACACCAGCCTAACCAGATTTCAAATTTTATCTTTTGG
## 6  CGGTATGCACTTTTAACAGTCACCCCCCAACTAACACATTATTTTCCCCTCCCACTCCCATACTACTAATCTCATCAATACA
## 7  ACCCCCGCCCATCCTACCCAGCACACACACACCGCTGCTAACCCCATACCCCGAACCAACCAAACCCCAAAGACACCCCCCA
## 8  CAGTTTATGTAGCTTACCTCCTCAAAGCAATACACTGAAAATGTTTAGACGGGCTCACATCACCCCATAAACAAATAGGTTT
## 9  GGTCCTAGCCTTTCTATTAGCTCTTAGTAAGATTACACATGCAAGCATCCCCGTTCCAGTGAGTTCACCCTCTAAATCACCA
## 10 CGATCAAAAGGGACAAGCATCAAGCACGCAGCAATGCAGCTCAAAACGCTTAGCCTAGCCACACCCCCACGGGAAACAGCAG
## 11 TGATTAACCTTTAGCAATAAACGAAAGTTTAACTAAGCTATACTAACCCCAGGGTTGGTCAATTTCGTGCCAGCCACCGCGG
## 12 TCACACGATTAACCCAAGTCAATAGAAGCCGGCGTAAAGAGTGTTTTAGATCACCCCCTCCCCAATAAAGCTAAAACTCACC
## 13 TGAGTTGTAAAAAACTCCAGTTGACACAAAATAGACTACGAAAGTGGCTTTAACATATCTGAACACACAATAGCTAAGACCC

Barchart example

  • Human mitochondrial DNA; 4 nucleotides (A,C,G,T); 16,568 bp

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Single quantitative variable

Histograms

  • We can visualise a single quantitative variable (continuous or discrete) using a histogram
  • First we need to bin our data: sort into non-overlapping intervals of equal size
  • The bins go on the x axis and the counts go on the y axis.

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Difference between barcharts and histograms

  • Typically barcharts will have spaces between the bars
  • In histograms, the bars are always touching
  • Barchart x axes are categorical
  • Histogram x axes are quantitative (typically have units)

Barchart

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Histogram

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Histogram example

  • Heights of 898 people, in metres 
##    height
## 1    1.86
## 2    1.76
## 3    1.75
## 4    1.75
## 5    1.87
## 6    1.84
## 7    1.66
## 8    1.66
## 9    1.80
## 10   1.73
## 11   1.79
## 12   1.74
## 13   1.70
  • Binned data
##    start finish counts
## 1   1.40   1.45      3
## 2   1.45   1.50      3
## 3   1.50   1.55     44
## 4   1.55   1.60    122
## 5   1.60   1.65    163
## 6   1.65   1.70    171
## 7   1.70   1.75    149
## 8   1.75   1.80    146
## 9   1.80   1.85     76
## 10  1.85   1.90     13
## 11  1.90   1.95      6
## 12  1.95   2.00      1
## 13  2.00   2.05      1

Histogram example

  • Using 14 bins

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Effect of bin size

  • Using 7 bins

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Effect of bin size

  • Using 28 bins

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Data distribution

Data distribution

  • The shape of the histogram can be called a distribution
  • Height is an example of the normal distribution
  • It looks (more or less) like a symmetrical bell
  • Very tall and very short people are rare, most people are around the middle
  • Many other variables: birth weight, blood pressure, measurement error

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Describing distributions for quantitative variables

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Mean, median, and mode

An average

  • An average describes the central position of the distribution
  • It summarises the entire distribution in a single value
  • We use three types:
    • mean
    • median
    • mode

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How are they calculated?

Mean

  • Add all values together and divide by number of observations
  • Good for continuous quantitative variables, discrete variables may need rounding

Median

  • Arrange values from smallest to largest and pick the middle one
  • Good for ordinal and quantitative continuous and discrete variables

Mode

  • The most frequently occurring value
  • Good for ordinal and discrete quantitative variables

Quantitative discrete variable example

1  2  3  3  4  4  4  5  5  5  5  6  6  6  7  7  8  9 10

Mean

1+2+3+3+4+4+4+5+5+5+5+6+6+6+7+7+8+9+10 = 100
100 / 19 = 5.263158

Median

5

Mode

5

Continuous variable example

  • Height of 500 women, plotted as a histogram

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mean = 1.651378
median = 1.6475637

  • Mean and median are virtually identical
  • A feature of the normal distribution
  • So why do we need different measures?
  • Not all variables are normally distributed

Ancient DNA fragment length

  • DNA in ancient samples is highly fragmented
  • The fragment lengths have a skewed distribution

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800 year old cat (Carl Vivian, Uni Leicester)

mean = 45.8054171
median = 41
mode = 31

Sample variability

An average doesn't give the whole picture

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Sample variability

  • Variability describes how narrow or spread out the distribution is
  • Also called dispersion, scatter or spread
  • We will look at two ways of measuring variability:

Range

  • The maximum minus the minimum values

Standard deviation

  • More complex
  • Positive number in same units as the data
  • Bigger numbers indicate higher variability

Range

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  • Mean = 15.0149312
  • Range = 27.6507856 - 3.017839 = 24.6329467

Standard deviation

  • Complex mathematical formula, the value is easy to understand
  • In research science, we would just use a computer to calculate!
  • square root (sum of squared differences from mean, divided by sample size minus 1)
Human Height Mean Difference Squared
1 160 165.33 -5.33 28.44
2 166 165.33 0.67 0.44
3 168 165.33 2.67 7.11
4 161 165.33 -4.33 18.78
5 167 165.33 1.67 2.78
6 170 165.33 4.67 21.78
TOTAL 79.33
  • SD = square root (sum of squares / 6 - 1) = square root (79.33/5) = 3.98

Or just use a computer

heights <- c(160, 166, 168, 161, 167, 170)
mean(heights)

## [1] 165.3333

sd(heights)

## [1] 3.983298

Now isn't that easier :)

Standard deviation and the normal distribution

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  • Mean = 15.0149312, Standard deviation = 3.0303673

Standard deviation and the normal distribution

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  • Mean = 15.0149312, Standard deviation = 3.0303673
  • 68% of observations fall within 1 SD of the mean, 95% with 2 SDs, and 99.7% within 3 SDs

Data analysis 2

  • Data visualisation
  • Single categorical or ordinal variable: barcharts
  • Single quantitative variable: histograms
  • Data distributions
  • Mean, median, and mode
  • Sample variability: range and standard deviation

Next time

Analysis of two variables, and hypothesis testing