Descriptive Statistics: Mean, Median Mode and Standard Deviation.

1. Data Science
Descriptive Statistics
(Mean, Median, Mode, Standard Deviation)

2. Mean
In statistics, mean is the most common and frequently used method to
measure the center of a data set. It’s a fundamental yet essential part of the
statistical analysis of data.
The mean (average) of a data set is found by adding all numbers in the data
set and then dividing by the number of values in the set.
Mean= Sum of observation / Total number of observation

3. Example
Find the mean of the following data set. 10, 20, 36, 12, 35, 40, 36, 30, 36,
40
• Mean = ∑xi/n
• = (10 + 20 + 36 + 12 + 35 + 40 + 36 + 30 + 36 + 40) /10
• = 295/10
• = 29.5
• Therefore, the mean of the given data set is 29.5.

4. Example- Grouped Data
Marks 25 43 38 42 33 28 29 20
Number of students 20 1 4 2 15 24 28 6
Mean = (∑fixi)/ ∑fi

5. Example- Grouped Data
Marks (xi) Number of students (fi) fixi
25 20 500
43 1 43
38 4 152
42 2 84
33 15 495
28 24 672
29 28 812
20 6 120
Sum 100 2878

6. Continue…
• Mean = (∑fixi)/ ∑fi
• = 2878/100
• = 28.78
• Thus, the mean of the given distribution is 28.78.

7. Median
In statistics, the median is a measure of central tendency, specifically a
measure of the middle value of a dataset when it's arranged in ascending or
descending order. The median is less sensitive to extreme values (outliers)
compared to the mean, making it a useful measure of central tendency,
especially when the data set contains outliers or is skewed.
Steps:
1. Arrange the data in ascending order (from smallest to largest) or
descending order (from largest to smallest).
2. If the number of data points is odd, the median is the middle value in the
ordered list.
3. If the number of data points is even, the median is the average of the two
middle values.

8. Example
• For example, consider the dataset: 3,6,9,12,15.
• Since there are 5 data points (an odd number), the median is the middle
value, which is 9.
• Consider the dataset: 2,4,6,8.
• Since there are 4 data points (an even number), the median is the average
of the two middle values, which is (4+6)/2=5.

9. Mode
In statistics, the mode is the value that appears most frequently in a dataset.
Unlike the mean and median, which are measures of central tendency, the
mode is a measure of the data's "typical" value based on frequency.
1. Identify the frequency of each unique value in the dataset.
2. Determine which value has the highest frequency. This value is the
mode.
A dataset can have one mode (unimodal), two modes (bimodal), or more
than two modes (multimodal). It's also possible for a dataset to have no
mode if all values occur with the same frequency.

10. Example
• Consider the dataset: 2,3,4,4,6,6,6,9.
• In this dataset, the value 6 appears most frequently (three times), so 6 is
the mode.
• Consider the dataset: 1,2,3,3,4,4,5.
• In this dataset, both 3 and 4 appear most frequently (twice each), so this
dataset is bimodal, with modes of 3 and 4.

11. Standard Deviation
The standard deviation is defined as the deviation of the values or data from
an average mean. Lower standard deviation concludes that the values are
very close to their average. Whereas higher values mean the values are far
from the mean value.
Standard Deviation is of two types:
1. Population Standard Deviation:
It measures the dispersion or spread of the entire population.
2. Sample Standard Deviation:
It estimates the population standard deviation based on the sample.

12. Formula for S.D
• σ = Standard Deviation
• xi = Terms Given in the Data
• μ = population mean
• x
̄ = Sample mean
• n = Total number of Terms
The formula for sample standard deviation
involves a correction for the fact that it's based
on a sample rather than the entire population.
The denominator in the formula is adjusted by
dividing by 𝑛−1 instead of n, where 𝑛 is the
number of data points in the sample. This
correction is known as Bessel's correction.

13. Example
During a survey, 6 students were asked how many hours per day they study
on an average? Their answers were as follows: 2, 6, 5, 3, 2, 3. Evaluate the
standard deviation.
• Find the mean of the data:
• (2+6+5+3+2+3)/6
• = 3.5
Mean =3.5

14. Construct the table
x1 x1 − x̄ (x1 − x̄)2
2 -1.5 2.25
6 2.5 6.25
5 1.5 2.25
3 -0.5 0.25
2 -1.5 2.25
3 -0.5 0.25
= 13.5
Mean=3.5

15. Use the Standard Deviation formula
• Sample Standard Deviation =
• 𝑠=√∑(𝑋−𝑋¯) 2 /𝑛−1
• =√(13.5/[6-1])
• =√[2.7]
• =1.643

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