How to calculate normal distributions in R

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Normal distribution is a continuous random variable distribution with bell-shaped probability density curve. It is widely used in statistical data analysis, and the basis for many other distributions as well. The probability density function can be expressed as

In which the two parameters μ and σ, are its mean and standard deviation, respectively.

The probability under the curve between any two x values x = x1 and x = x2 equals the integral

With R programming, there are generally following functions that are used to calculate for Normal distribution.

dnorm() – probability density

pnorm() – cumulative probability up to a specified value

qnorm() – quantile value, below which is the specified cumulative probability, and this is the opposite operation of pnorm()

rnorm() – to generate random number from a determined normal distribution

In the next code examples, you can see how to implement normal distributions with these functions.

# creating a sequence of values 
# between -3 to 3 with a step of 0.1
T = seq(-3, 3, by=0.1)
T
#output
[1] -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8
[14] -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5
[27] -0.4 -0.3 -0.2 -0.1  0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8
[40]  0.9  1.0  1.1  1.2  1.3  1.4  1.5  1.6  1.7  1.8  1.9  2.0  2.1
[53]  2.2  2.3  2.4  2.5  2.6  2.7  2.8  2.9  3.0
#calculate probability density for these values
#from a normal distribution with mean 0.5 standard deviation 1.2
dt = dnorm(T, mean=0.5, sd=1.2)
dt
#output
[1] 0.004725734 0.006005083 0.007577969 0.009496655 0.011818778
 [6] 0.014606917 0.017927867 0.021851574 0.026449710 0.031793853
[11] 0.037953294 0.044992472 0.052968089 0.061925970 0.071897766
[16] 0.082897616 0.094918912 0.107931330 0.121878295 0.136675062
[21] 0.152207571 0.168332238 0.184876796 0.201642270 0.218406128
[26] 0.234926563 0.250947860 0.266206671 0.280439019 0.293387772
[31] 0.304810305 0.314486023 0.322223431 0.327866430 0.331299555
[36] 0.332451900 0.331299555 0.327866430 0.322223431 0.314486023
[41] 0.304810305 0.293387772 0.280439019 0.266206671 0.250947860
[46] 0.234926563 0.218406128 0.201642270 0.184876796 0.168332238
[51] 0.152207571 0.136675062 0.121878295 0.107931330 0.094918912
[56] 0.082897616 0.071897766 0.061925970 0.052968089 0.044992472
[61] 0.037953294
# Plot the graph.
plot(T, dt)

#Cumulative probabilities for these values
#from a normal distribution with mean 0.5 standard deviation 1.2
PT <- pnorm(T, mean = 0.5, sd = 1.2)
PT
#output
[1] 0.001768968 0.002303266 0.002979763 0.003830381 0.004892537
 [6] 0.006209665 0.007831677 0.009815329 0.012224473 0.015130140
[11] 0.018610425 0.022750132 0.027640146 0.033376508 0.040059157
[16] 0.047790352 0.056672755 0.066807201 0.078290204 0.091211220
[21] 0.105649774 0.121672505 0.139330247 0.158655254 0.179658669
[26] 0.202328381 0.226627352 0.252492538 0.279834464 0.308537539
[31] 0.338461120 0.369441340 0.401293674 0.433816167 0.466793248
[36] 0.500000000 0.533206752 0.566183833 0.598706326 0.630558660
[41] 0.661538880 0.691462461 0.720165536 0.747507462 0.773372648
[46] 0.797671619 0.820341331 0.841344746 0.860669753 0.878327495
[51] 0.894350226 0.908788780 0.921709796 0.933192799 0.943327245
[56] 0.952209648 0.959940843 0.966623492 0.972359854 0.977249868
[61] 0.981389575
# Plot the graph for cumulative probabilities
plot(T, PT)

# Create a sequence of values representing cumulative probabilities 
prob_vec <- seq(0, 1, by = 0.05)
prob_vec
#output
[1] 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
[14] 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
#quantile values associated with these cumulative probabilities
#from a normal distribution with mean 0.5 standard deviation 1.2
QT <- qnorm(prob_vec, mean=0.5, sd=1.2)
QT
[1]        -Inf -1.47382435 -1.03786188 -0.74372007 -0.50994548
 [6] -0.30938770 -0.12928062  0.03761544  0.19598348  0.34920638
[11]  0.50000000  0.65079362  0.80401652  0.96238456  1.12928062
[16]  1.30938770  1.50994548  1.74372007  2.03786188  2.47382435
[21]         Inf
# Plot the graph for quantile values
plot(prob_vec, QT)

# Randomly generate 10000 numbers from a normal distribution
# with mean=0.5 and standard deviation=1.2
RT <- rnorm(10000, mean=0.5, sd=1.2)
# Plot generated random numbers in a histogram, with 50 bins
hist(RT, breaks=50)

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