Comprehensive Overview of Measures of Deviation and Dispersion in Statistics

Comprehensive Overview of Measures of Deviation and Dispersion in Statistics

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  RM &STAT. BAMS;FINAL PROF,2026 1 Measures of Deviation / Dispersion / Variability Measures of Deviation / Dispersion / Variability Measures of dispersion- It describe how widely the data values are spread around a central value (mean, median, or mode). They indicate the consistency or variability of a dataset. It helps to find out how individual observation value are dispersed from mean or difference between sample value to population value. 1. Range-The range is the difference between the highest and the lowest value in a dataset. Formula: Range = Maximum value − Minimum value Advantages:  It is the simplest measure of dispersion and Simple to calculate.  Gives a quick idea of variability.  It defines the normal limit of a biological characteristic. Eg, fasting blood sugar- 8- 110 mg/dl. Urea- 15-40 mg/dl; uric acid- 2-5mg/dl. Disadvantages:  Depends only on two extreme values., ignoring the distribution of all other values.  Highly affected by outliers. 2. Quartile Deviation (Semi-Interquartile Range)- Quartile deviation measures the spread of the middle 50% of observations. Formula: QD = (Q₃ − Q₁) / 2 Q₁ = First quartile Q₃ = Third quartile Advantages:  Less affected by extreme values.  Suitable for skewed distributions. Disadvantages: Ignores 50% of the observations. 3. Mean Deviation- Mean deviation is the average of the absolute deviations of observations from a central value (mean or median). Formula: Mean Deviation = Σ|X − A| / n X = Observation A = Mean or Median n = Number of observations Advantages:
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  RM &STAT. BAMS;FINAL PROF,2026 2 Measures of Deviation / Dispersion / Variability Uses all observations. Easy to understand. Disadvantages: Absolute values make mathematical calculations difficult. 4. Standard Deviation (SD)- Standard deviation is the square root of the variance and measures the average spread of observations around the mean. [Formula: SD = √Variance] Standard deviation is a measure of dispersion and most commonly used in' statistical analysis. The concept of SD was first introduced Karl Peacon in 1893. by SD is the positive square root, of the average of squared deviation taken from mean. The greater the value of SD, the further the data tend to be dispersed from the mean. Uses-  It is widely used in biological studies.  Most commonly used to measure of dispersion.  SD indicates whether the variation of difference of an individual from the mean. is by chance or real.  Helps in finding the standard error (SE)  Helps in finding the suitable size of sample for valid conclusions.  SD used to calculate Z-core (Relative deviation) and coefficient of variation, correlation.  The shape of normal curve will depend on mean and SD. In standard normal curve mean= 0; SD=1. Advantages:  Uses all observations.  Most reliable and widely used measure of variability.  Useful in statistical analysis. Disadvantages:  More difficult to calculate manually.  It cannot be used for qualitative data. Calculation of SD- 1. First calculate the mean of Series. Mean= Sum of observation/ member of observation. 2 . Find the difference (deviation) of individual measurement from mean. 3.Next, find the sum of the square of deviation of individual measurement from their mean. 4 Now, find the variance (var), which is mean square deviation. Var =(x-x)2/ η
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  RM &STAT. BAMS;FINAL PROF,2026 3 Measures of Deviation / Dispersion / Variability 5.Square root of variance. is SD. Example-find the SD of a math test score of five Students. 92, 88, 80, 68, 52, 1). Find the mean- 92+88+ 80 +68 +52 = 76 2). Find the deviation from. mean. 92-76=16 88-76=12 80-76=4 68-76= -8 52-76= -24 3) Square the deviation from mean (16)2 = 256; (12)2=144 ; (4)2 =16; (-8) 2 =64; (24)2 = 576. 4.) Find the sum of squares of the deviation from mem-256 + 144 + 16 + 64 +576 = 1056 5) Divided by the number of observations to find the variance. 1056/5 = 2111.2 6) Find the square root of variance is SD 2√var= √211.2=19.53 5. Variance and Coefficient of Variation (CV)- Variance- Variance is the average of the squared deviations from the mean. Formula: Variance = Σ(X − X ̄ )² / n (population) or Σ(X − X ̄ )² / (n − 1) (sample) Coefficient of Variation (CV)- CV is the standard deviation expressed as a percentage of the mean. It is used to compare variability between datasets. Formula: CV = (SD / Mean) × 100% Uses:  Compares variability between different datasets.  Lower CV indicates greater consistency. 6. Standard Error (SE)-  Standard error measures the variability of the sample mean from the true population mean. Formula: SE = SD / √n SD = Standard deviation n = Sample size  Chance variation from sample to sample or sample to population. Is measured by Standard error.  The Standard Error (SE) is the standard deviation of the sampling distribution of a statistical mean. ### Use SD, when we are talking about distribution, either of a sample or a population. We use SE , when we are talking about estimate found from sample.
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  RM &STAT. BAMS;FINAL PROF,2026 4 Measures of Deviation / Dispersion / Variability Uses:  Indicates the precision of the sample mean.  Used in confidence intervals and hypothesis testing.  It provides an idea about the unreliability of a sample  SE help to determining the limits within which the values are expected to lie, ie, confidence limit).  Chance variation from sample to sample or from sample to population is measured by SE.  To determine whether the sample is drawn from a known population or not. if Sample mean is larger than population mean, then 95% chances are the sample is not drowned from same population.  Help to calculate sample size, in order confidence limits. Standard Error of mean -Whatever be the sampling procedure, the Sample statistic will differ from the population parameter because of Chance or biological variability. Such difference between Statistics and parameter is measured by sampling error of mean. SE of mean is calculated by - = SD / √n n= sample size. So, SE varies directly with SD and inversely with the square root of the sample size. Standard Error of Proportion (SEP)- SEP is used as a measured of variation Which occur by chance in the proportion of a character from sample to sample or from sample to population. -It is used for qualitative data. It is frequently used to find the efficacy. of a drug, line of treatment; Operation, vaccine etc. SEP is calculated by.- SEP = √p. q / n P= percentage of positive character q= percentage of negative Character n= number of samples. Qualities of a Good Measure of Variability- A good measure of variability should: 1. Be rigidly and clearly defined. 2. Be easy to understand and calculate. 3. Be based on all observations. 4. Not be unduly affected by extreme values. 5. Be suitable for further mathematical and statistical analysis. 6. Show minimum sampling fluctuations. 7. Allow easy comparison between different datasets.