Normal Distribution in Statistics
You learnt about probability distributions of continuous random variables and understood how they differ from those of discrete random variables.
The most commonly occurring continuous probability distribution is a normal distribution. Several natural phenomena, such as the height of people of a certain age, blood pressure, and IQ scores, follow the normal distribution. Let’s learn about normal distribution in greater detail in the upcoming video.
You learnt that a normal distribution is symmetric about its mean and extends infinitely on both sides. In a normal distribution, the probability density is higher close to the mean and decreases exponentially as we move further away from the mean. In simple language, it means that there is a high probability that the value of the random variable is close to the mean. As we move further away from the mean, the probability of the occurrence of such values decreases.
Let’s take an example. Suppose a factory manufactures bolts that need to be of a certain diameter, say 10 mm. Now, if you measure the exact diameter of the bolts, you will notice that none of the bolts will be of exactly 10 mm diameter. However, most of the bolts will be about 10 mm in diameter. Also, it would be quite unlikely to find a bolt with a diameter of 15 mm if the process is working properly. By now, you must have guessed that the diameter of the manufactured bolts is a random variable that follows a normal distribution with a mean of 10 mm.
Apart from the mean, a normal distribution is defined by one more parameter – the standard deviation. The standard deviation provides a measure of dispersion/spread of the values of the normal distribution from the mean. Hence, the greater the standard deviation, the broader is the normal distribution.
Next, you learnt about the empirical rule which is illustrated in the following image:
The empirical rule states that there is a:
68% probability of the variable lying within 1 standard deviation of the mean,
95% probability of the variable lying within 2 standard deviations of the mean, and
99.7% probability of the variable lying within 3 standard deviations of the mean.
In other words, if you look at the outcome of one trial at random, there is a 68% chance that the outcome lies within one standard deviation of the mean, 95% chance that the outcome lies within two standard deviations of the mean, and 99.7% chance that the outcome lies within three standard deviations of the mean.
It is important to note that the numbers specified in the empirical rule are only approximations.
In the next segment, you will learn about a type of normal distribution that has the mean as 0 and standard deviation as 1.