## Dispersion in statistics.

Statistical calculation that allows us to tell how far apart our data is spread is called dispersion.

Variance is a simple measure of dispersion. Variance measures how far each number in the dataset from the mean. To compute variance first, calculate the mean and squared deviations from a mean.

Standard deviation is a squared root of the variance to get original values. Low standard deviation indicates data points close to mean.

## Computing the dispersion:


# import libraries
import matplotlib.pyplot as plt
import numpy as np

## create some data distributions

# the distributions
N = 10001   # number of data points
nbins = 30  # number of histogram bins

d1 = np.random.randn(N) - 1
d2 = 3*np.random.randn(N)
d3 = np.random.randn(N) + 1

# need their histograms
y1,x1 = np.histogram(d1,nbins)
x1 = (x1[1:]+x1[:-1])/2

y2,x2 = np.histogram(d2,nbins)
x2 = (x2[1:]+x2[:-1])/2

y3,x3 = np.histogram(d3,nbins)
x3 = (x3[1:]+x3[:-1])/2

meanval = 10.2
stdval  = 7.5
numsamp = 123

np.random.normal(meanval,stdval,numsamp)

# compute the means
mean_d1 = sum(d1) / len(d1)
mean_d2 = np.mean(d2)
mean_d3 = np.mean(d3)

## now for the standard deviation

# initialize
stds = np.zeros(3)

# compute standard deviations
stds = np.std(d1,ddof=1)
stds = np.std(d2,ddof=1)
stds = np.std(d3,ddof=1)

# same plot as earlier
plt.plot(x1,y1,'b', x2,y2,'r', x3,y3,'k')
plt.plot([mean_d1,mean_d1],[0,max(y1)],'b--', [mean_d2,mean_d2],[0,max(y2)],'r--',[mean_d3,mean_d3],[0,max(y3)],'k--')

plt.plot([mean_d1-stds,mean_d1+stds],[.4*max(y1),.4*max(y1)],'b',linewidth=10)
plt.plot([mean_d2-stds,mean_d2+stds],[.5*max(y2),.5*max(y2)],'r',linewidth=10)
plt.plot([mean_d3-stds,mean_d3+stds],[.6*max(y3),.6*max(y3)],'k',linewidth=10)

plt.xlabel('Data values')
plt.ylabel('Data counts')
plt.show()


## Computing different variance measures:

Mean absolute deviation of given data poins is the average distance between each data value and the mean. Mean absolute deviation is a way to describe variation in a data and gives us understanding of how "spread out" the values in a data set are.


variances = np.arange(1,11)
N = 300

varmeasures = np.zeros((4,len(variances)))

for i in range(len(variances)):

# create data and mean-center
data = np.random.randn(N) * variances[i]
datacent = data - np.mean(data)

# variance
varmeasures[0,i] = sum(datacent**2) / (N-1)

# "biased" variance
varmeasures[1,i] = sum(datacent**2) / N

# standard deviation
varmeasures[2,i] = np.sqrt( sum(datacent**2) / (N-1) )

varmeasures[3,i] = sum(abs(datacent)) / (N-1)

# show them!
plt.plot(variances,varmeasures.T)
plt.show()


## Computing Fano factor and coefficient of variation (CV):

Fano factor is a measure of the dispersion of a probability distribution of a Fano noise and might be considered to be an intensity independent measure of variability.

The coefficient of variation (CV) is a statistical measure of the dispersion of data set around the mean. In finance, the coefficient of variation allows investors to determine volatility or risk in comparison to the amount of return expected from investments.


# need positive-valued data (why?)
data = np.random.poisson(3,300)  # "Poisson noise"

fig,ax = plt.subplots(2,1)
ax.plot(data,'s')
ax.set_title('Poisson noise')

## compute fano factor and CV for a range of lambda parameters

# list of parameters
lambdas = np.linspace(1,12,15)

# initialize output vectors
fano = np.zeros(len(lambdas))
cv   = np.zeros(len(lambdas))

for li in range(len(lambdas)):

# generate new data
data = np.random.poisson(lambdas[li],1000)

# compute the metrics
cv[li]   = np.std(data) / np.mean(data) # need ddof=1 here?
fano[li] = np.var(data) / np.mean(data)

# and plot
plt.plot(lambdas,cv,'bs-')
plt.plot(lambdas,fano,'ro-')
plt.legend(('CV','Fano'))
plt.xlabel('$\lambda$')
plt.ylabel('CV or Fano')
plt.show()