Biometry: Mean, Mode & Median, standard deviation, co-efficient of variation, skewness and kurtosis

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:-

ii. If the numbers x1, x2,…, xk occur f1, f2,…,fk times respectively, the arithmetic mean is:-

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:

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:

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.

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.

If x1, x2,…,xk occur with frequencies f1, f2,…,fk respectively, the standard deviation can be computed as

Co-efficient of variation:- 

In statistic, the Coefficient of variation formula (CV), also known as relative standard deviation (RSD), is a standardized measure of the dispersion of a probability distribution or frequency distribution. When the value of the coefficient of variation is lower, it means the data has less variability and high stability.

The formula for coefficient of variation is given below:

σ = Standard deviation of the population

S = Standard deviation of the sample

μ = Mean

Skewness and Kurtosis:-

Skewness:- It is an important statistical technique that helps to determine asymmetrical behavior than of the frequency distribution, or more precisely, the lack of symmetry of tails both left and right of the frequency curve. A distribution or dataset is symmetric if it looks the same to the left and right of the center point.

Types of skewness:-

1. Symmetric Skewness:- A perfect symmetric distribution is one in which frequency distribution is the same on the sides of the center point of the frequency curve. In this, Mean = Median = Mode. There is no skewness in a perfectly symmetrical distribution.

2. Asymmetric Skewness:- A asymmetrical or skewed distribution is one in which the spread of the frequencies is different on both the sides of the center point or the frequency curve is more stretched towards one side or value of Mean. Median and Mode falls at different points.

i. Positive Skewness:- In this, the concentration of frequencies is more towards higher values of the variable i.e. the right tail is longer than the left tail.

ii. Negative Skewness:- In this, the concentration of frequencies is more towards the lower values of the variable i.e. the left tail is longer than the right tail.

Kurtosis:- It is also a characteristic of the frequency distribution. It gives an idea about the shape of a frequency distribution. Basically, the measure of kurtosis is the extent to which a frequency distribution is peaked in comparison with a normal curve. It is the degree of peakedness of a distribution. 

Types of kurtosis:-

1. Leptokurtic:- Leptokurtic is a curve having a high peak than the normal distribution. In this curve, there is too much concentration of items near the central value.

2. Mesokurtic:- Mesokurtic is a curve having a normal peak than the normal curve. In this curve, there is equal distribution of items around the central value.

3. Platykurtic:- Platykurtic is a curve having a low peak than the normal curve is called platykurtic. In this curve, there is less concentration of items around the central value.