MicrobiologyBytes: Maths & Computers for Biologists: Introduction to Statistics

Updated: February 6, 2009

Introduction to Statistics

Further information on this topic can be found in Chapter 7 of:

Numerical ability is an essential skill for everyone studying the biological sciences but many students are frightened by the 'perceived' difficulty of mathematics, and are nervous about applying mathematical skills in their chosen field of study. Maths from Scratch for Biologists is a highly instructive, informal text that explains step by step how and why you need to tackle maths within the biological sciences. (Amazon.co.UK)

Statistics (the systematic collection and display of numerical data) is the most abused area of numeracy.

97% of statistics are made up on the spot.

Unless appropriate statistical techniques are used, the results are meaningless:

Garbage In, Garbage Out!

The most important aspect of statistics is:

Choosing the right statistical test !

Statistics are a tool, not an objective in themselves - therefore framing the question to be asked in a form appropriate for statistical analysis is a critical step.

Statistics are not a mysterious black box - you must UNDERSTAND what you are trying to do you will not be able to construct the right questions and apply the right techniques:

Research
Question

Statistical
Question

Data
Collection

Statistical
Conclusion

Research
Conclusion

It is not sufficient to know the formulae for various statistical tests - you must know when (and when not) to apply them!

Statistical Variables:

Different classes of information are known as the variables of a dataset, e.g:

Age
Weight
Height
Gender
Marital status
Annual income

Variables which are experimentally manipulated by an investigator are called independent variables.
Variables which are measured are called dependent variables.
All other factors which may affect the dependent variable are called confounding, extraneous or secondary variables - unless these are the same for each group being tested comparisons will be unreliable.

Variables may be classified as either quantitative or qualitative:

Quantitative data measures either how much or how many of something, i.e. a set of observations where any single observation is a number that represents an amount or a count.
Qualitative data provide labels, or names, for categories of like items, i.e. a set of observations where any single observation is a word or code that represents a class or category.

Which of the above variables are quantitative and which are qualitative?

Qualitative data can be divided into:

Nominal variables: Variables with no inherent order or ranking sequence, e.g. numbers used as names (group 1, group 2...), gender, etc.

Ordinal variables: Variables with an ordered series, e.g. "greatly dislike, moderately dislike, indifferent, moderately like, greatly like". Numbers assigned to such variables indicate rank order only - the "distance" between the numbers has no meaning.

Interval variables: Equally spaced variables, e.g. temperature. The difference between a temperature of 66 degrees and 67 degrees is taken to be the same as the difference between 76 degrees and 77 degrees. Interval variables do not have a true zero, e.g. 88 degrees is not necessarily double the temperature of 44 degrees.

Ratio variables: Variables spaced equal intervals with a true zero point, e.g. age.

Quantitative data can be divided into:

Discrete variables: The set of all possible values which consists only of isolated points, e.g. counting variables (1, 2, 3 ...).

Continuous variables: The set of all values which consists of intervals, e.g. 0-9, 10-19, 20-29... etc.

variables

Statistical Methods:

Descriptive statistics are used to analyze the basic features of data under consideration.

Tabular methods: can be used to represent data sets with two variables, e.g:

	Age:
Gender:	20-29:	30-39:	40-49:	50-59:	60-69:
Male:	12	13	7	8	7
Female:	12	14	10	9	11

Numerical summaries:
- Proportion
- Percentage
- Mean (average)
- Median
- Mode
- Percentiles
- Range
- Variance
- Standard deviation
- etc.

Graphical methods: visual methods which summarize data, e.g:

Bar Charts / Histograms:	Pie Charts:	Scatter Diagrams:

plus many others... (look up "Chart Types" in MSExcel Help)

Frequency Distributions

The basis of most statistical investigations is construction of a frequency distribution:

the number of observations for each of the possible categories in a dataset

For nominal variables, the order in which the categories are listed makes no difference.
Ordinal, interval, ratio variables and quantitative data, categories should be listed in increasing rank order, e.g:

Age:	Frequency:
1	11
2	12
3	19
4	6
5	2

Grouped frequency distributions:

Remember that the aim of descriptive statistics is to analyze the basic features of a dataset. Complex data can be simplified by combining individual scores to form a smaller number of groups, referred to as class intervals. The, intervals should be:

Mutually exclusive - i.e. non-overlapping
All of the same width
Continuous throughout the distribution

e.g:

Marks:	Frequency:
91-100	1
81-90	2
71-80	9
61-70	0
51-60	2

Cumulative frequency distributions:

Cumulative frequency distributions are useful to show what proportion of a dataset lies above or below certain limits, e.g:

Marks:	Frequency:	Cumulative f	Cumulative %
91-100	1	50	100
81-90	9	49	98
71-80	9	40	80
61-70	13	31	62
51-60	7	18	36
41-50	6	11	22
31-40	4	5	10
21-30	0	1	2
11-20	0	1	2
1-10	1	1	2
	N = 50

What percentage of this class scored the required passmark of 41%?

Percentiles are points on a frequency distribution below which a specified percentage of cases in the distribution fall, e.g: a person scoring at the 75th percentile did better than 75% of those in the distribution:

(n+1)P

100

where:
n = number of cases
P = desired percentile

Marks:	Cumulative f	Cumulative %	Percentile:
95.27	129	100
94.13	128	99
92.46	127	98
91.66	126	98
91.66	125	97
91.59	124	96
91.46	123	95
90.92	122	95
90.60	121	94
90.06	120	93
89.66	119	92
89.52	118	91
89.39	117	91
89.06	116	90
88.59	115	89
87.99	114	88
87.93	113	88
87.39	112	87
87.26	111	86
87.12	110	85
86.99	109	84
86.93	108	84
86.86	107	83
86.72	106	82
86.59	105	81
86.59	104	81
86.53	103	80
86.52	102	79
85.99	101	78
85.86	100	78
85.66	99	77
85.19	98	76	P₇₅
85.12	97	75
84.92	96	74
84.86	95	74
84.80	94	73
84.52	93	72
84.39	92	71
84.32	91	71
83.92	90	70
83.65	89	69
83.52	88	68
83.39	87	67
82.92	86	67
82.92	85	66
82.79	84	65
82.59	83	64
82.59	82	64
82.52	81	63
82.46	80	62
81.85	79	61
81.72	78	60
81.66	77	60
81.66	76	59
81.58	75	58
81.19	74	57
80.85	73	57
80.79	72	56
80.79	71	55
80.72	70	54
80.12	69	53
80.05	68	53
79.99	67	52
79.66	66	51	P₅₀
79.52	65	50

Marks:	Cumulative f	Cumulative %	Percentile:
79.32	64	50
79.19	63	49
79.12	62	48
79.12	61	47
78.79	60	47
78.18	59	46
78.12	58	45
78.12	57	44
77.99	56	43
77.98	55	43
77.85	54	42
77.79	53	41
77.58	52	40
77.32	51	40
77.18	50	39
76.98	49	38
76.85	48	37
76.45	47	36
76.45	46	36
76.25	45	35
76.19	44	34
76.13	43	33
75.98	42	33
75.45	41	32
75.32	40	31
75.32	39	30
75.19	38	29
75.12	37	29
75.12	36	28
74.78	35	27
74.78	34	26
71.05	33	26	P₂₅
70.98	32	25
70.78	31	24
69.51	30	23
69.05	29	22
68.91	28	22
68.05	27	21
66.31	26	20
66.11	25	19
65.71	24	19
65.38	23	18
63.51	22	17
62.37	21	16
61.71	20	16
60.38	19	15
58.64	18	14
58.32	17	13
57.91	16	12
55.84	15	12
55.04	14	11
54.37	13	10
54.17	12	9
51.84	11	9
51.84	10	8
51.17	9	7
50.97	8	6
50.84	7	5
46.64	6	5
38.30	5	4
37.83	4	3
37.02	3	2
23.96	2	2
17.35	1	1

N.B. the 25th, 50th and 75th percentiles are also referred to as quartiles (Q).

To calculate percentiles from grouped data an interpolation method is required:

Find the class within which the percentile lies.
Determine the percentage between the bottom of the distribution and the class containing the percentile.
Determine the number of additional cases required to make up the percentile.
Assume (!) the scores in the class are evenly distributed.
Find the additional number of cases in the class required to make up the percentile.
Add this to the number of cases between the bottom of the distribution and the class containing the percentile.

P_x = LCB + (N/C * C)

where:
LCB = the value (percentage) of the lower class (frequency group) boundary
N = the number of additional cases required to make up the desired percentile
C = the number of cases in the class (frequency group)

e.g:

Marks:	Frequency:	Cumulative f	Cumulative %	Examples:
91-100	1	50	100	90% of 50 = 45 cases P₉₀ = 81 + (5/9 * 9) = 86%
81-90	9	49	98
71-80	9	40	80
61-70	13	31	61	50% of 50 = 25 cases P₅₀ = 61 + (7/13 * 9) = 65.85%
51-60	7	18	36
41-50	6	11	22
31-40	4	5	10
21-30	0	1	2
11-20	0	1	2
1-10	1	1	2
	N = 50

You may wish to note that MSExcel uses the following formulae to calculate quartiles. If the data are assumed to be in ascending order:

The lower quartile (Q₁) is the 1é4(n+3)th observation
The second quartile (Q₂ = median) is the 1é2(n+1)th observation
The upper quartile (Q₃) is the 1é4(3n+1)th observation
When 1é4(n+3), 1é2(n+1) and 1é4(3n+1) are not integers, linear interpolation is used.

(Freund, J and Perles, B (1987) "A New Look at Quartiles of Ungrouped Data", The American Statistician, 41, 3, 200-203)

Frequency Graphs

Remember that statistics is the science of collecting, classifying and interpreting numerical data. Graphs make it easy to see features in numerical data.

Histograms

Marks:	f	Cum f	Cum %
90-99	0	50	100
80-89	2	50	100
70-79	6	48	96
60-69	9	42	84
50-59	12	33	66
40-49	10	21	42
30-39	7	11	22
20-29	3	4	8
10-19	1	1	2
0-9	0	0	0

Frequency polygon

A frequency histogram is a series of rectangles representing the frequencies of the class intervals.

The same data can also be drawn as a frequency polygon:

Which format is best?

A histogram is best used for ungrouped data for a discrete variables.
A polygon is best for grouped data and continuous variables.

A number of differently-shaped frequency distributions are common:

Frequency distributions

In skewed (i.e. non-symmetrical) frequency distributions, the "tail" of the frequency distribution points in the direction of the skew, e.g.

Negative skew

Positive skew

In numerical terms, the shape of a frequency distribution is measured by two parameters:

Skew: The skew of a distribution can be calculated from the equation:

where: where µ is the mean and s is the standard deviation. A negative value indicates a negative skew and a positive value a positive skew. In Microsoft Excel, skew can be calculated using the SKEW function: SKEW(array).

Kurtosis: Based on the relative size of a frequency distribution's tails, kurtosis can be calculated from the equation:

where: where µ is the mean and s is the standard deviation. Kurtosis characterizes the relative peakedness or flatness of a distribution compared with the normal distribution (value = 0). Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis indicates a relatively flat distribution. In Microsoft Excel, kurtosis can be calculated using the KURT function: KURT(array).

The Normal (Gaussian) Distribution:

When many independent random factors act in an additive manner to create variability, the dataset follows a bell-shaped distribution called the normal (or Gaussian, after Carl Friedrich Gauss, 1777-1855) distribution:

Normal distribution

The normal distribution has some special mathematical properties which form the basis of many statistical tests.

Although no real datasets follow the normal distribution exactly, many kinds of data follow a distribution that is approximately Gaussian.

A normal distribution can be defined by 2 parameters, the mean and the standard deviation.

N.B: Not all datasets follow a normal distributions, e.g.

Binomial distribution: A probability distribution of binary variables. A binary variable is one which can take one of two possible values, e.g. male or female, left or right-handed, heads or tails, etc - the two states of a binary variable are mutually exclusive. Binary variables are quite common in biological data.
Poisson distribution: The Poisson Distribution is a discrete probability distribution that applies to occurrences of some event over a specific interval. The random variable X is the number of occurrences of the event in an interval. The interval can be time, distance, area, volume or some similar unit. This method can be used to model many events, e.g. the number of orchids in a field or the probability of your computer crashing.
etc ...

Parametric & Nonparametric Methods:

Statistical methods which depend on the parameters of populations or probability distributions and are referred to as parametric methods. Parametric tests include:

t-test
ANOVA
Regression
Correlation

These tests are only meaningful for continuous data which is sampled from a population with an underlying normal distribution or whose distribution can be rendered normal by mathematical transformation.

Nonparametric methods require fewer assumptions about a population or probability distribution and are applicable in a wider range of situations:

Methods used with qualitative data.
or:
Methods used with quantitative data when no assumption can be made about the population probability distribution.

Nonparametric methods are useful in situations where the assumptions required by parametric methods appear questionable. A few of the more commonly used nonparametric methods include:

These tests are characterised as distribution free - i.e. neither the values obtained nor the population from which the sample was drawn need have a normal distribution. Unlike the parametric tests which can give erroneous results these can always be used safely regardless of the distribution of the data.

Unfortunately, they are less flexible in practice and less powerful than parametric tests. In cases where both parametric and nonparametric methods are applicable, statisticians usually recommend using parametric methods because they tend to provide better precision.

Accuracy and Precision:

Accuracy - a measurement of how close the average of a large set of measurements is to the true or target value.
Precision - a measure of the closeness of repeated observations to each other without rererence to the true or target value, i.e. the reproducibility of the result.

Consider the results of the Aardvark's Archery Club:

Accuracy and precision

Ann Archer: Good precision (grouping) and accuracy (bull's-eye!)
Bill Tell: Good precision, poor accuracy
Quentin Quiver: Reasonable accuracy, poor precision
King Harold: Poor accuracy and precision

Choosing an Appropriate Statistical Test

In order to choose the appropriate statistical test, you must answer two questions:

What kind of dataset is being analyzed?
What is the goal of the analysis?

Goal:	Dataset:
Goal:	Measurement (from a normal distribution)	Rank, Score, or Measurement (from non-normal distribution)	Binomial (e.g. heads or tails)	Survival Time:
Describe one group:	Mean, SD	Median, interquartile range	Proportion	Kaplan-Meier survival curve
Compare one group to a hypothetical value:	One-sample t test	Wilcoxon test	Chi-square or Binomial test
Compare two unpaired groups:	Unpaired t test	Mann-Whitney test	Fisher's exact test (or chi-square for large samples)	Log-rank test or Mantel-Haenszel
Compare two paired groups:	Paired t test	Wilcoxon test	McNemar's test	Conditional proportional hazards regression
Compare three or more unmatched groups:	One-way ANOVA	Kruskal-Wallis test	Chi-square test	Cox proportional hazard regression
Compare three or more matched groups:	Repeated-measures ANOVA	Friedman test	Cochrane Q test	Conditional proportional hazards regression
Quantify association between two variables:	Pearson correlation	Spearman correlation	Contingency coefficients
Predict value from another measured variable:	Simple regression	Nonparametric regression	Simple logistic regression	Cox proportional hazard regression
Predict value from several measured or binomial variables:	Multiple regression		Multiple logistic regression	Cox proportional hazard regression

On this course, we will consider only the tests in the shaded cells.
For information on other statistical tests, consult the table above.

So how do you choose an appropriate statistical test?

We will investigate this further in the next session, Descriptive Statistics.