Question

1. What is the link between the normal distribution and (Variation 2 (and egg roulette)), where we relied on statements such as “about two-thirds of observations are within 1 standard deviation of the mean”?

2. Do natural phenomena such as hemoglobin levels or the weight of ants really follow a normal distribution?

3. If you add up a large number of random events, you get a normal distribution. How large a number makes a normal distribution? 4. Are the results of studies due to adding up a large number of random events?

Answer 1

1. Yes, there is a link between the normal distribution and the statements such as “about two-third of observations are within 1 standard deviation of the mean”.

The statement such as “about two-third of observations are within 1 standard deviation of the mean” means that

P(Mean-Standard deviation<Observations< Mean+ Standard deviation) = 2/3 = 0.67

Also, in the normal distribution,

P(Mean-Standard deviation<Normal variable X< Mean+ Standard deviation) =0.68(approximately)

Since, the above probabilities are almost equal then, there is a link between the normal distribution and the statements such as “about two-third of observations are within 1 standard deviation of the mean”..

2. The distribution of hemoglobin levels of women is significantly different from normal distribution. It is a negatively skewed distribution. Also, the transformation to the square of the hemoglobin levels also shows significant skewness, but the extent of departure of the transformation from the normal distribution is not significant.(Reference: Distribution of haemoglobin level, packed cell volume, and mean corpuscular haemoglobin concentration in women in the community by by p. c. Elwood* Belfast, Brit. J. prev. soc. Med. (1964), 18, 81-87).

By considering the hemoglobin levels of both men and women, the hemoglobin levels in general may hardly approximately follow normal distribution but not exactly.

We know that human weights which are measured in kgs follow normal distribution. Similar to weights of human beings, weights of ants are also measured and shown in milligrams. Therefore, the distribution weights of ants are not significantly different from normal distribution.

3. Usually, in most of the cases, the addition of 30 or more than 30 random number of events follows approximately normal distribution.

4. According to central limit theorem, due to adding up of large number of events, the addition follows normal distribution.