Advance Statistics Learning Topics -Jobelle M Quilana Section 7001-1-1-B.pptx

Master of Arts in Education - Major in Educational Management
JOBELLE M. QUILANA
Section 7001-1-1-B
ADVANCED
STATISTICS

LEARNING TOPICS:
 Summation Notation
 Measures of Central Tendency
 Arithmetic Mean
 Median

 Summation notation (or sigma notation) allows
us to write a long sum in a single expression.
 This is the sigma symbol: Σ. It tells us that we
are summing up something.

Summation: A series of addition
𝐸𝑥𝑎𝑚𝑝𝑙𝑒:
𝑖=1
𝑛
𝑥𝑖
read as “the summation of 𝑥 sub 𝑖, 𝑖 is from
1 to 𝑛.
𝑖=1
𝑛
𝑥𝑖 = 𝑥1 + 𝑥2 + 𝑥3 + ⋯ + 𝑥𝑛

Summation
 This means taking the sum of 𝑛 number of observations or
values of the variable represented by 𝑥.
 The subscript 𝑖 represents the order of an observation, whether it
is the first, second, third or the last.
 The notation 𝑖 = 1 under the summation sign Σ, denotes the
lower limit and indicates the start of counting.
 The number above, 𝑛, is the upper limit and tells the total number
of observations to be added.

Summation
Thus, the symbol
𝑖=1
3
𝑥𝑖
indicates the sum of the first three variables, while the symbol
𝑖=2
5
𝑥𝑖
indicates the sum of the second to fifth values of 𝑥.

Suppose the grades obtained by five high school students in a high school
mathematics test are as follows: 85, 76, 70, 80, and 75.
 There are five observations, hence 𝑛 = 5.
 These are five values of 𝑥 represented as 𝑥1 = 85, 𝑥2 = 76, 𝑥3 = 70, 𝑥4 = 80 and
𝑥5 = 75.
The symbol used to represent the sum of these five numbers would then be
𝑖=1
5
𝑥𝑖 = 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5
= 85 + 76 + 70 + 80 + 75
= 386.

Summation Rules
Summation of 𝑛 constants
𝑖=1
𝑛
𝑐𝑖 = 𝑛𝑐
1.
𝑖=1
5
7 = 7 + 7 + 7 + 7 + 7 = 5 7 = 35.

Summation Rules
Summation of a Sum
𝑖=1
𝑛
(𝑥𝑖 + 𝑦𝑖) = 𝑥1 + 𝑦1 + 𝑥2 + 𝑦2 + ⋯ + 𝑥𝑛 + 𝑦𝑛 =
𝑖=1
𝑛
𝑥𝑖 +
𝑖=1
𝑛
𝑦𝑖
1. = 𝑥1 + 𝑦1 + 𝑥2 + 𝑦2 + 𝑥3 + 𝑦3 =
𝑖=1
3
(𝑥𝑖 + 𝑦𝑖)
𝑖=1
3
𝑥𝑖 +
𝑖=1
3
𝑦𝑖

Summations Rules
Summation of a Variable and a Constant
𝑖=1
𝑛
(𝑥𝑖 + 𝑐) =
𝑖=1
𝑛
𝑥𝑖 +
𝑖=1
𝑛
𝑐
𝐸𝑥𝑎𝑚𝑝𝑙𝑒𝑠:
1.
𝑖=1
4
(𝑥𝑖 + 4)
2.
𝑖=3
6
(𝑥𝑖 − 3)
𝑖=1
4
𝑥𝑖 +
𝑖=1
4
4
=
=
𝑖=3
6
𝑥𝑖 −
𝑖=3
6
3

Summation Rules
Sum of a Product
𝑖=1
𝑛
(𝑥𝑖)(𝑦𝑖) = 𝑥1 𝑦1 + 𝑥2 𝑦2 + ⋯ + (𝑥𝑛)(𝑦𝑛)
1.
𝑖=1
3
(𝑥𝑖𝑦𝑖)
2.
𝑖=3
5
𝑥𝑖𝑦𝑖
= 𝑥1 𝑦1 + 𝑥2 𝑦2 + 𝑥3 𝑦3
= 𝑥3 𝑦3 + 𝑥4 𝑦4 + 𝑥5 𝑦5

Summation Rules
Summation of the Product of a Constant and a Variable
𝑖=1
𝑛
𝑐𝑥𝑖 = 𝑐
𝑖=1
𝑛
𝑥𝑖
1.
𝑖=1
4
10𝑥𝑖
2.
𝑖=3
5
−5𝑥𝑖
𝑖=1
4
𝑥𝑖
𝑖=3
5
𝑥𝑖
= 10
= −5

Summation Rule
Square of the Sum of Variables
𝑖=1
𝑛
𝑥𝑖
2
= 𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛
2
1.
𝑖=1
5
𝑦𝑖
2
2.
𝑖=3
7
𝑥𝑖𝑦𝑖
2
= 𝑦1 + 𝑦2 + 𝑦3 + 𝑦4 + 𝑦5
2
= 𝑥3𝑦3 + 𝑥4𝑦4 + 𝑥5𝑦5 + 𝑥6𝑦6 + 𝑥7𝑦7
2

Summation Rule
Sum of the Squares of Variables
𝑖=1
𝑛
𝑥𝑖
2
= 𝑥1
2
+ 𝑥2
2
+ ⋯ + 𝑥𝑛
2
1.
𝑖=1
3
𝑥𝑖
2
2.
𝑖=3
5
𝑥𝑖𝑦𝑖
2
= 𝑥1
2
+ 𝑥2
2
+ 𝑥3
2
= 𝑥3𝑦3
2
+ 𝑥4𝑦4
2
+ 𝑥5𝑦5
2

MEASURES OF CENTRAL TENDENCY
 A measure of central tendency is a summary measure that attempts to
describe a whole set of data with a single value that represents the
middle or center of its distribution.
There are three main measures of central tendency:
 mean
 median
 mode

 The mean is the sum of the value of each observation in a dataset divided by
the number of observations. This is also known as the arithmetic average.
Example: Looking at the retirement age distribution again:
 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60
 The mean is calculated by adding together all the values
(54+54+54+55+56+57+57+58+58+60+60 = 623) and dividing by the number
of observations (11) which equals 56.6 years.
Mean

Median
 The median is the middle value in distribution when the values are arranged
in ascending or descending order.
The median divides the distribution in half (there are 50% of observations on
either side of the median value).
 In a distribution with an odd number of observations, the median value is the
middle value.
 When the distribution has an even number of observations, the median value
is the mean of the two middle values.

Median
Example:
Looking at the retirement age distribution below (which has 11 observations), the
median is the middle value, which is 57 years.
54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60
In the distribution below, the two middle values are 56 and 57, therefore the
median equals 56.5 years.
52, 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60

Mode
 The mode is the most commonly occurring value in a distribution.
Example: Consider this dataset showing the retirement age of 11 people, in whole
years: 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60
This table shows a simple frequency distribution of the retirement age data.
The most commonly occurring value is
54, therefore the mode of this
distribution is 54 years.

Weighted mean
 Used when some data values are more important than the
others.
Formula:
Weighted Mean =
Where:
x = data/each number in the data
w = weight/importance

Weighted mean
Example: Find Carl’s GPA using weighted mean
Subject Grade (x) Units (w)
English 4 3
Math 4 3
Physics 3 4
Statistics 2.33 3
Chemistry 2.67 2
Solution: Weighted Mean =
= (3x4) + (3x4) + (4x3) + (3x2.33) + (2x2.67)/3+3+4+3+2
= 48.33/15
Weighted Mean = 3.22

Midrange
 The value that is halfway between the minimum data value and the maximum data value.
Formula: Midrange = minimum value + maximum value
2
Example: Find the midrange of the following daily temperatures which were recorded at 3
hour interval.
52˚, 65˚, 71˚, 74˚, 76˚, 75˚, 68˚, 57˚, 54˚
Solution: first, get the minimum and maximum value from the given data. (min val = 52˚, max
val = 76˚)
Midrange = 52˚+76˚ = 128/2 = 64˚
2

Arithmetic Mean
 The arithmetic mean in statistics, is nothing but the ratio of all
observations to the total number of observations in a data set.
 Some of the examples include the average rainfall of a place, the
average income of employees in an organization. We often come
across statements like "the average monthly income of a family is
P15,000 or the average monthly rainfall of a place is 1000 mm"
quite often. Average is typically referred to as arithmetic mean.

Arithmetic Mean
 Arithmetic mean is often referred to as the
mean or arithmetic average. It is calculated
by adding all the numbers in a given data
set and then dividing it by the total number
of items within that set.
 The formula for calculating arithmetic
mean is (sum of all observations)/(number
of observations). For example, the
arithmetic mean of a set of numbers {10,
20, 30, 40} is (10 + 20 + 30 + 40)/4 = 25.

 The arithmetic mean of ungrouped data is calculated using the formula:
 Mean x
̄ = Sum of all observations / Number of observations
 Example: Compute the arithmetic mean of the first 6 odd natural numbers.
 Solution: The first 6 odd natural numbers: 1, 3, 5, 7, 9, 11
 x
̄ = (1+3+5+7+9+11) / 6 = 36/6 = 6.
Calculating Arithmetic Mean for Ungrouped Data

Calculating Arithmetic Mean for Grouped Data
 There are three methods to calculate the arithmetic mean for grouped
data.
1. Direct method
2. Short-cut method
3. Step-deviation method

Direct Method
 Let x₁, x₂, x₃ ……xₙ be the observations with the frequency f₁,
f₂, f₃ ……fₙ.
 Then, arithmetic mean is calculated using the formula:
 x
̄ = (x₁f₁+x₂f₂+......+xₙfₙ) / ∑fi
 Here, f₁+ f₂ + ....fₙ = ∑fi indicates the sum of all frequencies.

Direct Method
 Example I (discrete grouped data): Find the arithmetic mean of the following distribution:
x 10 30 50 70 89
f 7 8 10 15 10
xi fi xifi
10 7 10×7 = 70
30 8 30×8 = 240
50 10 50×10 = 500
70 15 70×15 = 1050
89 10 89×10 = 890
Total ∑fi=50 ∑xifi=2750
Add up all the (xifi) values to obtain ∑xifi. Add up all the fi
values to get ∑fi
Now, use the arithmetic mean formula.
x
̄ = ∑xifi / ∑fi = 2750/50 = 55
Arithmetic mean = 55.

Direct Method
 Example II (continuous class intervals): Let's try finding the mean of the following
distribution:
Class-Interval 15-25 25-35 35-45 45-55 55-65 65-75 75-85
Frequency 6 11 7 4 4 2 1
Solution: When the data is presented in the form of
class intervals, the mid-point of each class (also
called class mark) is considered for calculating the
arithmetic mean.
Note: Class Mark = (Upper limit + Lower limit) / 2
Class-
Interval
Class
Mark (xi)
Frequenc
y (fi)
xifi
15-25 20 6 120
25-35 30 11 330
35-45 40 7 280
45-55 50 4 200
55-65 60 4 240
65-75 70 2 140
75-85 80 1 80
Total 35 1390
x
̄ = ∑xifi/ ∑fi = 1390/35 = 39.71.
Arithmetic mean = 39.71

Short-cut Method
 The short-cut method is called as assumed mean method or change of
origin method. The following steps describe this method.
 Step1: Calculate the class marks (mid-point) of each class (xi).
 Step2: Let A denote the assumed mean of the data.
 Step3: Find deviation (di) = xi – A
 Step4: Use the formula:
 x
̄ = A + (∑fidi/∑fi)

Example: Calculate the mean of the following using the short-cut method.
Short-cut Method
Class-Intervals 45-50 50-55 55-60 60-65 65-70 70-75 75-80
Frequency 5 8 30 25 14 12 6
Solution: Let us make the calculation table. Let the assumed mean be A =
62.5
Note: A is chosen from the xi values. Usually, the value which is around the middle is taken.
Class- Interval
Classmark/
Mid-points (xi)
fi di = (xi - A) fidi
45-50 47.5 5 47.5-62.5 =-15 -75
50-55 52.5 8 52.5-62.5 =-10 -80
55-60 57.5 30 57.5-62.5 =-5 -150
60-65 62.5 25 62.5-62.5 =0 0
65-70 67.5 14 67.5-62.5 =5 70
70-75 72.5 12 72.5-62.5 =10 120
75-80 77.5 6 77.5-62.5 =15 90
∑fi=100 ∑fidi= -25
Now we use the formula,
x
̄ = A + (∑fidi/∑fi) = 62.5 + (−25/100) =
62.5 − 0.25
= 62.25

Step Deviation Method
 This is also called the change of origin or scale method. The
following steps describe this method:
 Step 1: Calculate the class marks of each class (xi).
 Step 2: Let A denote the assumed mean of the data.
 Step 3: Find ui = (xi−A)/h, where h is the class size.
 Step 4: Use the formula to find the arithmetic mean:
 x
̄ = A + h × (∑fiui/∑fi)

Step Deviation Method
Class Intervals 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Total
Frequency 4 4 7 10 12 8 5 50
Example: Consider the following example to understand this method. Find the arithmetic mean of the
following using the step-deviation method.
Solution: To find the mean, we first have to find the class marks and decide A (assumed mean).
Let A = 35 Here h (class width) = 10
C.I. xi fi ui= xi−Ahxi−Ah fiui
0-10 5 4 -3 4 x (-3)=-12
10-20 15 4 -2 4 x (-2)=-8
20-30 25 7 -1 7 x (-1)=-7
30-40 35 10 0 10 x 0= 0
40-50 45 12 1 12 x 1=12
50-60 55 8 2 8 x 2=16
60-70 65 5 3 5 x 3=15
Total ∑fi=50 ∑fiui=16
Using arithmetic mean formula:
x
̄ = A + h × (∑fiui/∑fi) =35 + (16/50) ×10 =
35 + 3.2 = 38.2

Median
 Median, in statistics, is the middle value of the given list of data when
arranged in an order. The arrangement of data or observations can be
made either in ascending order or descending order.
 The median of a set of data is the middlemost number or center value in
the set.
 The median is also the number that is halfway into the set.

Median formula
 Odd Number of Observations
If the total number of observations (n) given is odd, then the formula to calculate the
median is:
 Even Number of Observations
If the total number of observations (n) is even, then the median formula is:

Median formula
Example 1: Imagine that a top running athlete in a typical 200-metre training session runs
in the following times: 26.1 seconds, 25.6 seconds, 25.7 seconds, 25.2 seconds, 25.0
seconds, 27.8 seconds and 24.1 seconds. How would you calculate his median time?
Solution: Let’s start with arranging the values in increasing order:
Rank Times (in
seconds)
1 24.1
2 25.0
3 25.2
4 25.6
5 25.7
6 26.1
7 27.8
There are n = 7 data points, which is an uneven
number. The median will be the value of the data
points of rank
(n + 1) ÷ 2 = (7 + 1) ÷ 2 = 4.
The median time is 25.6 seconds.

Median formula
If the number of data points is even, the median will be the average of the data point of rank n ÷ 2
and the data point of rank (n ÷ 2) + 1.
Example 2: Now suppose that the athlete runs his eighth 200-metre run with a time of 24.7 seconds.
What is his median time now?
Solution: Let’s start with arranging the values in increasing order:
Rank Times (in
seconds)
1 24.1
2 24.7
3 25.0
4 25.2
5 25.6
6 25.7
7 26.1
8 27.8
There are now n = 8 data points, an even number. The median is
the mean between the data point of rank
n ÷ 2 = 8 ÷ 2 = 4
and the data point of rank
(n ÷ 2) + 1 = (8 ÷ 2) +1 = 5
Therefore, the median time is (25.2 + 25.6) ÷ 2 = 25.4 seconds.

References
 https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/review-summation-
notation
 https://www.scribd.com/presentation/340528254/21-4-Summation-Notation-%CE%A3
 https://www.abs.gov.au/statistics/understanding-statistics/statistical-terms-and-concepts/measures-
central-tendency
 https://www.youtube.com/watch?v=dlwvRDibAd8
 https://www.cuemath.com/data/arithmetic-mean/
 https://byjus.com/maths/median/
 https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch11/median-mediane/5214872-eng.htm

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