# Chapter 5 Statistics

## Describing a Single Set of Data

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In [1]:

num_friends = [100,49,41,40,25,21,21,19,19,18,18,16,15,15,15,15,14,14,13,13,13,13,12,12,11,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,8,8,8,8,8,8,8,8,8,8,8,8,8,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]

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In [2]:

from collections import Counter

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In [3]:

%matplotlib inline
from matplotlib import pyplot as plt

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In [4]:

# Figure 5-1. A histogram of friend counts
friend_counts = Counter(num_friends)
xs = range(101)
ys = [friend_counts[x] for x in xs]
plt.bar(xs, ys)
plt.axis([0, 101, 0, 25])
plt.title("Histogram of Friend Counts")
plt.xlabel('# of friends')
plt.ylabel('# of people')
plt.show()

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In [5]:

num_points = len(num_friends)
num_points

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Out[5]:

204

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In [6]:

largest_value = max(num_friends)
largest_value

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Out[6]:

100

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In [7]:

smallest_value = min(num_friends)
smallest_value

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Out[7]:

1

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In [8]:

sorted_values = sorted(num_friends)

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In [9]:

smallest_value = sorted_values[0]
smallest_value

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Out[9]:

1

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In [10]:

second_smallest_value = sorted_values[1]
second_smallest_value

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Out[10]:

1

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In [11]:

second_largest_value = sorted_values[-2]
second_largest_value

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Out[11]:

49

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## Central Tendencies

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In [12]:

def mean(x):
return sum(x) / len(x)

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In [13]:

mean(num_friends)

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Out[13]:

7.333333333333333

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In [14]:

def median(v):
"""finds the middle-most value"""
n = len(v)
sorted_v = sorted(v)
midpoint = n // 2

if n % 2 == 1:
# if odd, return the middle value
return sorted_v[midpoint]
else:
# if even, return the average of middle values
lo = midpoint - 1
hi = midpoint
return (sorted_v[lo] + sorted_v[hi]) / 2

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In [15]:

median(num_friends)

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Out[15]:

6.0

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In [17]:

def quantile(x, p):
"""returns the pth-percentile value in x"""
p_index = int(p * len(x))
return sorted(x)[p_index]

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In [18]:

quantile(num_friends, 0.10)

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Out[18]:

1

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In [19]:

quantile(num_friends, 0.25)

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Out[19]:

3

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In [20]:

quantile(num_friends, 0.70)

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Out[20]:

8

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In [22]:

quantile(num_friends, 0.90)

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Out[22]:

13

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In [23]:

def mode(x):
"""returns a list, might be more than one mode"""
counts = Counter(x)
max_count = max(counts.values())
return [x_i for x_i, count in counts.items() if count == max_count]

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In [24]:

mode(num_friends)

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Out[24]:

[1, 6]

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## Dispersion

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In [25]:

def data_range(x):
return max(x) - min(x)

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In [26]:

data_range(num_friends)

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Out[26]:

99

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In [28]:

def de_mean(x):
"""translates x by substracting its mean"""
x_bar = mean(x)
return [x_i - x_bar for x_i in x]

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In [29]:

def variance(x):
"""assume x has at least 2 elements"""
n = len(x)
deviation = de_mean(x)
return sum(dev ** 2 for dev in deviation) / (n - 1)

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In [30]:

variance(num_friends)

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Out[30]:

81.54351395730716

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In [31]:

import math

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In [32]:

def standard_deviation(x):
return math.sqrt(variance(x))

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In [33]:

standard_deviation(num_friends)

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Out[33]:

9.03014473623248

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In [34]:

def interquartile_range(x):
return quantile(x, 0.75) - quantile(x, 0.25)

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In [35]:

interquartile_range(num_friends)

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Out[35]:

6

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## Correlation

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In [36]:

def covariance(x, y):
n = len(x)
return sum(x_i * y_i for x_i, y_i in zip(de_mean(x), de_mean(y))) / (n - 1)

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In [37]:

daily_minutes = [1,68.77,51.25,52.08,38.36,44.54,57.13,51.4,41.42,31.22,34.76,54.01,38.79,47.59,49.1,27.66,41.03,36.73,48.65,28.12,46.62,35.57,32.98,35,26.07,23.77,39.73,40.57,31.65,31.21,36.32,20.45,21.93,26.02,27.34,23.49,46.94,30.5,33.8,24.23,21.4,27.94,32.24,40.57,25.07,19.42,22.39,18.42,46.96,23.72,26.41,26.97,36.76,40.32,35.02,29.47,30.2,31,38.11,38.18,36.31,21.03,30.86,36.07,28.66,29.08,37.28,15.28,24.17,22.31,30.17,25.53,19.85,35.37,44.6,17.23,13.47,26.33,35.02,32.09,24.81,19.33,28.77,24.26,31.98,25.73,24.86,16.28,34.51,15.23,39.72,40.8,26.06,35.76,34.76,16.13,44.04,18.03,19.65,32.62,35.59,39.43,14.18,35.24,40.13,41.82,35.45,36.07,43.67,24.61,20.9,21.9,18.79,27.61,27.21,26.61,29.77,20.59,27.53,13.82,33.2,25,33.1,36.65,18.63,14.87,22.2,36.81,25.53,24.62,26.25,18.21,28.08,19.42,29.79,32.8,35.99,28.32,27.79,35.88,29.06,36.28,14.1,36.63,37.49,26.9,18.58,38.48,24.48,18.95,33.55,14.24,29.04,32.51,25.63,22.22,19,32.73,15.16,13.9,27.2,32.01,29.27,33,13.74,20.42,27.32,18.23,35.35,28.48,9.08,24.62,20.12,35.26,19.92,31.02,16.49,12.16,30.7,31.22,34.65,13.13,27.51,33.2,31.57,14.1,33.42,17.44,10.12,24.42,9.82,23.39,30.93,15.03,21.67,31.09,33.29,22.61,26.89,23.48,8.38,27.81,32.35,23.84]

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In [39]:

covariance(num_friends, daily_minutes)

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Out[39]:

22.425435139573064

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In [40]:

def correlation(x, y):
stdev_x = standard_deviation(x)
stdev_y = standard_deviation(y)
if stdev_x > 0 and stdev_y > 0:
return covariance(x, y) / (stdev_x * stdev_y)
else:
return 0

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In [41]:

correlation(num_friends, daily_minutes)

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Out[41]:

0.24736957366478216

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In [42]:

outlier = num_friends.index(100)

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In [43]:

num_friends_good = [x
for i, x in enumerate(num_friends)
if i != outlier]

daily_minutes_good = [y
for i, y in enumerate(daily_minutes)
if i != outlier]

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In [44]:

correlation(num_friends_good, daily_minutes_good)

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Out[44]:

0.5736792115665573

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