Biometry: Mean, Mode & Median, standard deviation and SB experimental errors, hypothesis testing, reliability and validity of results and inferences from experiments
Biometry:- It is the statistical analysis of biological observations and phenomena. It other words the application of statistical analysis to biological data.
Measures of Central Tendency:- The central tendency of a distribution is an estimate of the "center" of a distribution of values. There are three major types of estimates of central tendency:
1. Mean
2. Median
3. Mode
1. Mean:- It is probably the most commonly used method of describing central tendency. To calculate the mean add up all the values and divide by the number of values.
i. If the numbers x1, x2,…, xk occur only once, the arithmetic mean is:-
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ii. If the numbers x1, x2,…, xk occur f1, f2,…,fk times respectively, the arithmetic mean is:-
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2. Median:- It is the score found at the exact middle of the set of values. One way to compute the
median is to list all scores in numerical order, and then locate the score in the center of the
sample. If the two
middle scores had different values, then average of two would determine the median.
For grouped data, the median is obtained using following:
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where:
L = lower class limit of the median class (i.e., the class containing the median),
(Σf)1 =
sum of frequencies of all classes lower than the median class,
fm = frequency of median class
c = class interval
3. Mode:- is the most frequently occurring value in the set of scores. To determine the mode,
order the scores and then count each one. The most frequently occurring value is the mode.
In case of grouped data, the mode will be the value (or values) of x corresponding to the
maximum point (or points) on the curve. From a frequency distribution or histogram, the mode
can be obtained from the formula:
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where L is the lower class limit of modal class (the class containing the mode), f1 is the
frequency of the class previous to the modal class, f2 is frequency of the class just after the modal
class and c is the size of modal class.
Measures of Dispersion:-
Dispersion:- The scatterness or variation of observations from their average is called the dispersion.
Range:- It is simply the highest value minus the lowest value.
Standard Deviation (SD):- It is
a more accurate and detailed estimate of dispersion because an outlier can greatly exaggerate the
range. The Standard Deviation shows the relation that set of scores has to the mean of the sample.
The standard deviation is the square root of the sum of the squared deviations from the mean
divided by the number of scores.
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If x1, x2,…,xk occur with frequencies f1, f2,…,fk respectively, the standard deviation can be
computed as
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Standard Error:- It is a mathematical tool used in statistics to estimate the variability. It is used to estimate the efficiency, accuracy, and consistency of a sample. In other words, it measures how precisely a sampling distribution represents a population.
Where:
SE = Standard Error
S = Standard Deviation
n = Number of samples
Hypothesis Testing:- The main purpose of statistics is to test a hypothesis.
i. Statistical Hypothesis:- It is an assumption either about the form or about the parameters of a
distribution. For example, average height of a particular species of tree is 50 feet, normal distribution has mean 20.
ii. Simple and composite hypothesis:- If all the parameters are completely specified, hypothesis is called a simple hypothesis, otherwise it is a composite hypothesis. For example, average height of tree is 50 feet is a simple hypothesis and average height of tree is greater than 50 feet is a composite hypothesis.
iii. Null Hypothesis (H0):- The hypothesis under test for a sample study is called Null hypothesis
(H0). It represents a theory that has been put forward, either because it is believed to be true or
because it is to be used as a basis for argument, but has not been proved. For example, in a
clinical trial of a new drug, null hypothesis might be that the new drug is, on average, as effective
as the current drug i.e. H0: Effect of the two drugs, on the average, is same.
iv. Alternative Hypothesis (H1):- Any null hypothesis is tested against a rival, which is called
Alternative hypothesis (H1).
Examples of tests:-
i. Chi square test
ii. Independent Samples t-Test
iii. Paired t-Test
iv. Goodness of Fit
v. Analysis of Variance
Reliability and validity of results:-
> Reliability is another term for consistency. If one person takes the samepersonality test several times and always receives the same results, the test isreliable.
> A test is valid if it measures what it is supposed to measure. If theresults of the personality test claimed that a very shy person was in factoutgoing, the test would be invalid.
> Reliability and validity are independent of each other. A measurement maybe valid but not reliable, or reliable but not valid.
> Suppose your bathroomscale was reset to read 10 pound lighter. The weight it reads will be reliable (the same every time you step on it) but will not be valid, since it is notreading your actual weight.
Inferences from experiments:-
> The thinking skill you need to develop is called inference.
> When you make an inference, you are trying to come up with the explanation that best fits your observations.
> When explaining something, ask yourself ‘Why did this happen?’ or ‘what is the way this may have happened?’
> An inference isn’t always a correct explanation, but it should fit the evidence.
> You should always be ready to change your inference as more observations come to hand.