1  Introduction

1.1 Scientific method and design of experiment

Process by which scientific information is collected, analyzed and reported in order to produce unbiased, replicable and reproducible results to provide an accurate representation of observation.

Steps to scientific method:

  1. Making an observation
  2. Formulating a hypothesis
  3. Designing an experiment
  4. Conclusion (which needs to be replicable/reproducible)

Accuracy: Correctness of measurements. It is also called Validity.

Precision: Consistency of measurements. It is also called Reliability.

Sample - statistics Population - parameters
Mean \(\bar{x}\) \(\mu\)
Standard Deviation s 𝝈
Variance s2 𝝈2
Size n N

Note: Random Sampling is difficult in biology/real world due to following reasons:

  1. Sampling units do not represent natural distinct habitats
  2. Cannot be numbered in advance
  3. Population covers a large area
  4. Hence haphazard sampling method are used which ar assumed to be same as random sampling

Random Sample ( More technical definition )

  1. Simple Random Sampling

    If a sample size n is drawn from a population of size N in such a way that every possible sample of size n has the same chance of being selected, the sample of size n has the same chance of being selected, the sample is called a random sample.

    1. with replacement: every individual member of population is available at each draw
    2. without replacement: one less( or more) individual after each draw

  2. Systematic Random Sampling:

    eg. health records, stored in a systematic way. Total number of records needed to be studied are calculated. A random number table is then employed to select the starting point in a file system.

    eg. if we have x as the starting point then x+k , x+2k, x+3k can be different record sets where k is an interval used to define the sampling interval. It is not useful for study where you observe a gradient.

    ID Data point
    1 24
    2 34
    3 36
    4 34
    5 27
    6 91
    7 16
    8 17
    9 29
    10 20
    11 21
    12 32

  3. Stratified Random Sampling: When sample units are grouped i.e. the population is partitioned into strata ( which are more similar in nature than other strata), much less variability is observed

    1. Stratified systematic sampling
    2. Stratified sampling proportional to size of population in each strata

1.2 Measures of Central Tendency:

Mean Median Mode
Sample Population
$\bar{x} = \frac{\sum_{i=1}^n}{n} $ \(\mu = \frac{\sum_{i=1}^N}{N}\)

odd: \((n+1)/2\)

even: \((n/2)*(n/1+1)\)

 Most frequent value
unique; simple; affected by extreme values unique; simple; not affected by extreme values

Skewness: \(\frac{\sqrt{n}\sum_{i=1}^n(x_i-\bar{x})^3}{\sum_{i=1}^n(x_i-\bar{x})^{3/2}}\)

Positive Skewness/ Right skewed/ Right tailed Negative Skewness/ Left skewed/ Left tailed
Mean > Mode Mean < Mode
skewness > 0 skewness < 0

Dispersion ~ variation ~ spread ~ scatter

Range= largest data point to smallest data point; \(R = x_L - x_S\)

Variance:

Sample Population
\(s^2=\frac{\sum_{i=1}^n(x_i-\bar{x})^{2}}{n-1}\) \(\sigma^2=\frac{\sum_{i=1}^N(x_i-\mu)^{2}}{N}\)

Coefficient of variations:

C.V = \(\frac{s}{x}* (100) \%\)

Percentiles and Quartiles:

Given a set of observations \(x_1\), \(x_2\), \(x_3\),…… \(x_n\), the \(p^{th}\) percentile \(p\) is the value of \(x\) such that \(p\) percent or less of the observations are less than \(p\) and \((100 -p)\) percent or less of observations are greater than \(p\).