This document provides an overview of key concepts in statistics including measures of central tendency (mean, median, mode), measures of variation (range, quartiles, variance, standard deviation), the normal distribution, percentiles, and problems involving calculating and comparing these concepts. It defines terms like mean, median, mode, range, variance, standard deviation, quartiles, box plots, percentiles, and the normal distribution. It also includes examples of calculating these measures from data sets and interpreting them.

1. Concept Session: Statistics
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2. Topics Measures of Central Location
• Mean
• Median
• Mode
Measures of Variation
• Range
• Quartiles
• Variance
• Standard Deviation
• Normal Distribution
• Percentiles
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3. What Is Central Tendency?
A score that indicates where the center of the distribution tends
to be located.
Basic measures of central tendency
Mean
Median
Mode
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4. Mean
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(Arithmetic)Mean of a set of data is given by
𝑆𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠
Example:
Find the mean of: 6, 8, 11, 5, 2, 9, 7, 8
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5. Mean
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A random sample of 25 women beyond child-bearing age
gave the following data, where x is the number of children
and f is the frequency of that value, the number of times it
occurred in the data set. What is the mean number of
children per woman in the sample?
x 0 1 2 3
f 4 14 5 2
Answer : 1.2
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6. Median
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The median value of a set of data is the middle value of the
ordered data (ascending/descending).
Example:
Find the median of the following:
a) 15, 3, 9, 7, 11, 5, 6
b) 1.8, 3.9, 0.7, 0.9, 2, 2.5, 3.1, 3.2
Answers :
a) 7
b) 2.25
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7. Median
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A random sample of 25 women beyond child-bearing age
gave the following data, where x is the number of children
and f is the frequency of that value, the number of times it
occurred in the data set. What is the median number of
children per woman in the sample?
x 0 1 2 3
f 4 14 5 2
Answer : 1
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8. Mode
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The modal value of a set of data is the most frequently occurring
value.
Example:
Find the mode for: 2, 6, 3, 9, 5, 6, 2, 6
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9. Mode
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A random sample of 25 women beyond child-bearing age
gave the following data, where x is the number of children
and f is the frequency of that value, the number of times it
occurred in the data set. What is the mode of the sample?
x 0 1 2 3
f 4 14 5 2
Answer : 1
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10. Range
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The range is strongly affected by outliers
Range = max  min
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11. Quartiles
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Three numbers which divide the ordered data into four
equal sized groups.
Q1 has 25% of the data below it.
Q2 has 50% of the data below it. (Median)
Q3 has 75% of the data below it.
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12. Quartiles
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Algorithm to find Quartiles
1. Order the data.
2. For Q2, just find the median.
3. For Q1, find the median of this lower half.
4. For Q3, find the median of this upper half.
Example:
Find the 1st & 3rd Quartiles, Q1 and Q3, of the following set of
numbers :
347, 242, 146, 391, 249, 567, 277, 218, 319
Answer :
Q1 = 230
Q3 = 369
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13. Box Plot
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14. Variance
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Each data value has an associated deviation from the mean:
Steps to find Variance
1. Find the mean
2. Find the deviation of each value from the mean
3. Square the deviations
4. Sum the squared deviations
5. Divide the sum by n-1
xxi 
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15. Standard Deviation(SD)
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𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
• small values of SD indicate small variability in the data
• large values of SD indicate large variability in the data
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16. The Normal Distribution
X
f(X)
Changing μ shifts
the distribution left
or right.
Changing σ increases or
decreases the spread.
µ
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17. Key Areas under the Curve
 For normal distributions
+ 1 SD ~ 68%
+ 2 SD ~ 95%
+ 3 SD ~ 99.9%
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18. Problem
Family income in a city is normally distributed with
mean $25000 and standard deviation of $10000. If the
poverty level is $15,000, what percentage of the
population lives in poverty?
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Ans: 16

19. Problem
Joe buys 6 pens with an average(arithmetic mean) price
of $15. If Joe buys 2 more pens with an
average(arithmetic mean) price of $20, what is the
average price all the pens taken together ( in $, correct to
two decimal places)
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Ans: $16.25
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20. Problem
The average ( arithmetic mean) of 21 consecutive
integers is 25, what is the value of the smallest
integer among the 21 integers?
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Ans: 15
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21. Problem
Dataset A: -12, -10, -8, -6
Dataset B: -12, -11, -10, -9
Dataset C: 12, 12, 12, 12
Dataset D: -12, -24, -36, -48
Arrange the datasets above in ascending order of standard deviation.
A. C, B, A, D
B. C, A, B, D
C. B, C, A, D
D. A, B, C, D
E. C, D, B, A
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Ans: A
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22. Problem
Given below is the sample of ages (in months) of 18 children at
a day care:
36, 42, 18, 32, 22, 22, 25, 29, 30, 31, 19, 24, 35, 29, 26, 36, 24,
28
The interquartile range for this data set is
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Ans : 8 ( 32-24)
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23. Problem
Rank the following measures in order or “least affected by
outliers” to “most affected by outliers”.
A. mean, median, range
B. median, mean, range
C. range, median, mean
D. median, range, mean
E. range, mean, median
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Ans: option B (median, mean, range)
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24. Problem
The values of 12 houses on a particular street, as estimated by a
real estate agency, are shown in the table.
Find the mean value of these houses in dollars.
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Ans: $225,000
Estimated value per house Number of houses
$100,000 1
$175,000 5
$200,000 4
$225,000 6
$700,000 1
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25. Problem
Quantity A Quantity B
The average(arithmetic mean)
weight of the 5 people in the
assignment group
The median weight of the 5
people in the assignment group
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An assignment group in a class consists of 5 students.
Weight of the heaviest student is 60% more than the
weight of the lightest student
Answer : Option D
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26. Multi-modal Normal Distribution
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27. Problems
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Which of the following would the data pattern shown best describe?
I. A number of male employees and a larger number of female employees have
normally distributed salaries, distributed around the same mean.
II. A number of students have normally distributed weights, and a smaller
number of heavier, adult teachers also have normally distributed weights.
III. The time taken to complete a 10km run for a number of male athletes are
normally distributed, and the corresponding time taken for a smaller number
of female athletes are also normally distributed, although around a smaller
mean.
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28. Percentile
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The nth percentile is the smallest score that is greater
than or equal to a certain percentage of the scores.
Scores Rank
3
5
7
8
9
11
13
15
1
2
3
4
5
6
7
8
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29. Percentile Rank
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Marks Rank
3
5
7
8
9
11
13
15
1
2
3
4
5
6
7
8
𝑅 =
𝑃
100
∗ (𝑁 + 1)
Where, R is the percentile Rank
P is the desired percentile
N is the number of values
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30. Percentile Rank
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Marks Rank
3
5
7
8
9
11
13
15
1
2
3
4
5
6
7
8
Find the 25th percentile mark in the
distribution given
Answer : 5.5
𝑅 =
25
100
∗ 8 + 1 =
9
4
= 2.25.
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31. Problem
Quantity A Quantity B
95th percentile score 150
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The 80th percentile on a test corresponds to 140 marks,
while the 40th percentile corresponds to a score of 70
marks.
Answer : Option D
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