Normal Distribution

Application example: the amount of time a user spends on a website, the height of all the students in a class.

• The mean and median and mode fall at the center of the ideal normal distribution
• Area under the curve is all the 1.0 and 100%
• Normal distribution follows empirical rule 68-95-99.7
  • 2sigma covers 68% of the data
  • 4sigma covers 95% of the data
  • 6sigma covers 99.7% of the data
  • little tails 0.03 % of the data and equally divided on both sides.
• Percentile 95th means 95% of the data lies below that curve
• Other forms of Gaussian is t-distribution/exponential distribution

Z-score

• How far is the value from the mean in terms of standard deviations? That is defined by z-score.
• $z= (X-u)/sigma$ u is mean, X is data point, sigma is std deviation.
• for ex: if mean is 16 and the value we are looking for is 17.5, and std is 1, (17.5-16) / 1 z-score is 1.5 then we look into the table it says this value z-table is 93.38% above all the values.
• If the value of z is 1.5 that means its 1.5 std above the mean and the area under the curve will always be the same.
• The z value and the z-table is used to identify the percentage of area under the curve, which is also the p-value
• z-score always give area left to the curve

z-statistic

• When population standard deviation is known we can use the z-score

  

• When population standard deviation is unknown and we have sample greater than 30 samples, we use z-statistic for proportions only if we can assume the population is normally distributed, we use sample standard deviation in the formula, that can be proven by

  

Probability Density Function

• It means how much probability is concentrated per unit length (d𝒙) near 𝒙, or how dense the probability is near 𝒙. x is any point on the x-axis and y is the P(x)

<aside> 💡 The probability Density curve is not the same as the Probability of a function at (X=x). It is the integral of the probability density function.

</aside>

Plots

Probability Density Function Plot

• norm.pdf returns a PDF value, we can use this function to plot the normal distribution function.
• The following is a PDF of the normal distribution using scipy, numpy and plotly.
• The domain of −10<𝑥<10, the range of 0<P(𝑥)<0.20, the default values 𝜇=0 and 𝜎=1

from scipy.stats import norm
import scipy.stats as stats
import plotly.graph_objs as go
import plotly.graph_objects as go
import plotly.graph_objects as go
from plotly.subplots import make_subplots
from plotly.offline import init_notebook_mode, plot_mpl
colors=['#151515','#f0c24f']
fig = go.Figure()
x = np.linspace(-10,10,1000)
**p = norm.pdf(x, scale=2)**
fig.add_trace(go.Scatter(y=p , x=x, mode='lines+markers',marker_color=colors[1]))
fig.update_layout(title="Probability Density Function mu=0, sigma=2", 
                 legend=dict(x=.05,y=0.95, traceorder='reversed', font_size=16), 
                 width=500,
                 height=500,
                 yaxis=dict(
                          title="Probability Density Function P(x)",
                 titlefont=dict(
                          color="#1f77b4"
                                ),
                 tickfont=dict(
                        color="#1f77b4"
                               )
  ))
fig.show()

Probability Density Function with varying standard deviation

• What does this really mean?
  • It means if we have a set of data say between 1000 linearly spaced points between (-10 to 10) and we plot a graph for constant mean =0 and keeping varying the sigma between [2,4,6].
  • The observation is the graph gets broader which means the probability density is spreading out.