BIOSTATISTICS-Normal Curve, Test of significance, Standard error

Negatively skewed

For a symmetric curve such as gaussian – mean, median and mode coincide. That is, they are the same value.

– A skewed distribution will have different mean, median and mode. In fact, very different values of mean, median and mode is a clear indication of skewness.

In positive skew, mean>median>mode. In negative skew, mean<median<mode.

A negatively skewed distribution is where most of the values are on the higher side and the tail is pointing left.

The mean of a distribution is more affected by extreme values than is the median.

Therefore, in a negatively skewed distribution with few values on the lower end of the distribution, the mean is pulled towards the tail or lower end and becomes less than the median.

Ref: Park’s Textbook Of Preventive And Social Medicine, By K. Park, 19th Edition, Pages 699-702; Essentials of Research Methods in Health, Physical Education, Exercise Science and Recreation, By Kris E. Berg, Richard W. Latin, Second Edition, Pages 85-87; High-Yield Biostatistics, By Anthony N. Glaser, Third Edition, Pages 10, 11.

• 67% of the observations lie between the mean ± 1 standard deviation
• 95% of the observations lie between the mean ± 2 standard deviations
• 99.7% of the observations lie between the mean ± 3 standard deviations

Ans. is ‘c’ i.e., Mean and standard deviation 

o Normal distribution curve is based on mean and standard deviation

Ans. is ‘c’ i.e., Bell shape 

o Standard normal curve (Gaussian distribution) is bell shape curve.

Ans. is ‘b’ i.e., Curve is bilaterally symmetrical 

o Confidence limit of normal curve can never be 100%; therefore, limbs of curve never touch the baseline.

o Mean, Median and mode all coincide —> No skew.

Ans. is ‘a’ i.e., Distribution of data is symmetrical 

o In standard normal curve data is distributed symmetrically on either side of central value.

o Mean = Median = mode = zero

Ans. is c i.e., 50%

Ans. is ‘a’ i.e., 95% of all children weight between 12 and 18 kg 

o Here the examinor himself has given us the valve of standard error = 1.5

o So, we can directly apply sample data to population.

o 95% of data will be covered by 2 standard error = 15 ± 2 SE =12 ± 3

= 12 to 18

Ans. is ‘a’ i.e., Dispersion; ‘c’ Variation

Standard error of mean

• It is simply referred to as standard error.

o If we measure a sample from a wider population, then the mean of the sample will be an approximation of the total population mean.

• But how accurate is this ?

o The answer of this question is to calculate standard error of mean : ‑

o If we take multiple samples from a wide population, and measure the mean of each sample, we will find that every sample has different mean.

o If we make a frequency distribution of all these means (means of every sample), we will find the distribution of these means is normal Gaussian distribution.

o The mean of all these sample means (mean of every sample) will be same as the population mean.

Standard error of mean is the standard deviation of the mean of sample means, and thus gives a measure of their spread.

Ans. is ‘a’ i.e., Increase with sample size increase 

Ans. is ‘a’ i.e., 0.4

Ans. is ‘d’ i.e., 0.1 

o Standard error of mean is calculated as Standard Deviation/ VT₁; where ‘n’ is the total number of values within the sample. Thus the standard error of mean hemoglobin from the above sample would be (1.0/ Vioo ) or 1/10 or 0.1.

Ans. is ‘a’ i.e., Less than median

• Sidedness of the skewed distribution is towards the side of tail. For example right sided skewed deviation means the tail is towards the right.

Facts to remember the relation between mean, media & mode (see above figure) :-

i)    Mean is right of the median under right skew, and left of the median under left skew.

ii)  Mode is left of the median under right skew, and right of the median under left skew.

Ans. is ‘b’ i.e., Mean < Mode 

Ans. is ‘b’ i.e., Skewed distribution 

Ans. D: Mean = Mode

In a normal distribution curve, mean, mode and median coincide

Normal distribution

• It describes real world situations based on study results.
• It is used for continuous quantitative variables.
• It has an infinite range.
• It is the distribution that is normally seen.
• Although it is called “Normal” it applies to most biomedical measurements specially with big number of observations.
• It is the most important tool in analysis of epidemiological and research data.
• Has a Bell Shape Curve and is Symmetric
• It is Symmetric around a central axis (the mean)
• The halves of the curve are the same (mirror images)
• Mean = Median = Mode determine the location of the curve
• The total area under the curve is 1 (or 100%)
• Distinguishing features of normal distribution
• The mean ± 1 standard deviation covers 68% of the area under the curve (68% of cases)
• The mean ± 2 standard deviation covers 95% of the area under the curve (95% of cases)
• The mean ± 3 standard deviation covers 99.7% of the area under the curve (almost all cases)

Ans. is ‘b’ i.e., Positively skewed

In question, mean > median > mode → Feature of positively skewed deviation.

Ans. is ‘b’ i.e., Mean > Medians > Mode

Ans:B.)Right Skewed Distribution

Normal Distribution Curve 

• A histogram is the graphical representation of quantitative data .
• It is used for continuous quantitative variables like systolic BP, height, weight etc.
• It has an infinite range.
• Characteristics of a Normal Curve
• It is bell shaped
• It is symmetrical bilaterally.
• Total area of the curve is 1; its mean is zero; and its standard deviation is 1.
• Mean, mode and median coincide and = 0
• It has two inflictions at the points where the curve changes from convexity to concavity
• Mean ± 1 SD includes 68.27% of observations
• Mean ± 2 SD will include 95% of the values and
• Mean ± 3 SD will include 99.5% of observations.

Skew distribution.
Skewed to left (negatively skewed)

• A left-skewed distribution has a long left tail. 
• The mean is also to the left of the peak.
• Median>mean.

Skewed to Right (Positively skewed)

• A right-skewed distribution has a long right tail.
•  The mean is also to the right of the peak.
• Median<mean.

Poisson distribution .

• Limiting form of binominal distribution when probability  of   success  is   closer   to   zero   and numbers of trials are infinite. Mean=variance.

Ans: B. 1-Normal distribution, 2-Negative skewed, 3-Positive skewed

Positive or negative skewed data:

• Defined by the direction of the tail, i.e. direction of the least frequency values.
• Similarly in this box plot, we can see that the data is equally distributed on either side of the mean box in Plot (1).
• In Plot (2), the median is towards the higher side and most values are distributed towards the higher side hence it is negatively skewed.
• Hence, vice versa for Plot (3).

Answer- A. 0.68

The area under the Normal curve with ± 1 SD is 0.68.