Question

What is the significance of measures of dispersion? How is standard deviation
calculated and interpreted? Give an example?​

Answer 1

Step-by-step explanation:

While measures of central tendency are used to estimate "normal" values of a dataset, measures of dispersion are important for describing the spread of the data, or its variation around a central value.

Standard deviation (represented by the symbol sigma, σ ) shows how much variation or dispersion exists from the average (mean), or expected value. More precisely, it is a measure of the average distance between the values of the data in the set and the mean.

Step 1: Find the mean. Step 2: For each data point, find the square of its distance to the mean. Step 3: Sum the values from Step 2. Step 4: Divide by the number of data points.

Answer 2

Answer:

The measure of dispersion indicates the scattering of data. It explains the disparity of data from one another, delivering a precise view of the distribution of data. The measure of dispersion displays and gives us an idea about the variation and central value of an individual item.

In other words, Dispersion is the extent to which values in a distribution differ from the average of the distribution. It gives us an idea about the extent to which individual items vary from one another and from the central value.

The variation can be measured in different numerical measures, namely:

(i) Range – It is the simplest method of measurement of dispersion and defines the difference between the largest and the smallest item in a given distribution. Suppose, If Y max and Y min are the two ultimate items then

Range = Y max – Y min

(ii) Quartile Deviation – It is known as Semi-Inter-Quartile Range, i.e. half of the difference between the upper quartile and lower quartile. The first quartile is derived as (Q), the middle digit (Q1) connects the least number with the median of the data. The median of a data set is the (Q2) second quartile. Lastly, the number connecting the largest number and the median is the third quartile (Q3). Quartile deviation can be calculated by

Q = ½ × (Q3 – Q1)

(iii) Mean Deviation-Mean deviation is the arithmetic mean (average) of deviations ⎜D⎜of observations from a central value {Mean or Median}.

Mean deviation can be evaluated by using the formula: A = 1⁄n [∑i|xi – A|]

(iv) Standard Deviation- Standard deviation is the Square Root of the Arithmetic Average of the squared of the deviations measured from the mean. The standard deviation is given as

σ = [(Σi (yi – ȳ) ⁄ n] ½ = [(Σ i yi 2 ⁄ n) – ȳ 2] ½

Apart from a numerical value, graphics method are also applied for estimating dispersion.

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