SM025 CHAPTER 6 - DATA DESCRIPTION

Note: 1 of 3 Mathematics 2 SM025
Ch. 6 Data Description

Topic: 6. DATA DESCRIPTION

Sub-Topic: 6.1 Introduction to Data

Learning Outcomes: At the end of this lesson, students should be able to
(a) identify the discrete and continuous data.

*Include: Quantitative and qualitative variables.

(b) identify ungrouped and grouped data.
(c) construct and interpret stem-and-leaf diagrams.

*Interpretation on the shape of the distribution, bell shaped, skewed to the right or skewed to the left.

Statistics in our life

 In the field of science, statistical techniques are used to analyze data that is created from
the experiment.

 In manufacturing, quality control is achieved with the aid of statistics.
 In the area of business, marketing surveys are carried out to determine the compatibility of

the product with the economics and social demand.
 In the formation of the national policy, data from the census is used in economic and social

planning.
 In the field of education, statistical techniques are used to analyze the progress of students

in an examination.

Definition

 STATISTICS is a science that deals with collecting, organizing, summarizing, presenting
and analyzing data.

 DESCRIPTIVE STATISTICS consists of techniques involving collecting, tabulating,
 presenting and summarizing information in clear and effective ways in order to describe the

set of data.
 POPULATION PARAMETER is a summary measure of a population (such as

population, means, variances, e.t.c.)
 A SAMPLE is a set of measurements that constitute part or all of a population, i.e., a

sample is a subset of a population.
 A VARIABLE is any measured characteristic or attribute that differs for different

subjects.
 For example, if the weight of 30 subjects were measured, then weight would be a variable.

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Note: 1 of 3 Mathematics 2 SM025
Ch. 6 Data Description

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 Quantitative variables are measured on an ordinal, interval, or ratio scale.
 Qualitative variables are measured on a nominal scale.

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EXAMPLE 1
A survey was carried out on 30 women and 50 men in Kedah to find out whether they support
the action taken by the government to revoke the licenses of small petrol kiosks near the Thai
border. State
(a) the population.

(b) the sample.

(c) the variable.

(d) the type of variable, qualitative or quantitative.

EXAMPLE 2

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Categorise the following information into qualitative or quantitative data.
(a) Height of boys.

(b) Type of footwear.

(c) Age of lecturers.

(d) Colour of cars.

(e) Make of motorcycles.

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(f) Number of A’s scored in a public examination.

EXAMPLE 3
Based on the following statements, determine either the data is discrete data or continuous data.
(a) The time taken to travel from Ipoh to Kuala Lumpur.

(b) The number of pens sold by a stationary shop.

(c) The diameter of ten spheres.

(d) The number of customers in a cinema in one day.

(e) The weight of new born babies in a hospital.

Introduction of Ungrouped and Grouped Data

UNGROUPED DATA GROUPED DATA

Ungrouped Data are listed as a sequence Grouped Data is grouped in interval, are categorized into

or in the form of a frequency table but mutually exclusive intervals, can be presented in

without the use of intervals. frequency distribution table, histogram, polygon, ogive.

(a) Sequence: Weight(kg) 30  40 40  50 50  60 60  70

12,13, 21, 27, 33, 34, 35, 37, 40, 40, 41 Frequency 2 8 6 5
(b) Frequency table:

Number of children 0 1 2 3 4
Number of families 4 6 7 2 1

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X

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Ungrouped Data

Stem-and-Leaf Diagrams

 A stem and leaf diagram will contain all of your data in all of its detail.
 Also known as Stem Plots.
 The shape of stem and leaf is the same as the histogram.
 Each value in the data is divided into two parts as the stem and the leaf.
 If all the data are two-digit numbers, the first digit is the stem and the second digit is the leaf.
 For example, 58 can be divided as 5 8 where 5 is the stem and 8 is the leaf.

EXAMPLE 4
The length of 16 leaves of a certain tree, correct to the nearest 0.1 cm are given below.

4.4 5.9 6.0 5.7 6.2 5.3 8.0 5.0 6.3 5.7 6.6 5.5 4.3 5.9 4.8 8.1.
Construct a stem and leaf diagram to represent these figures.

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Note: 1 of 3 Mathematics 2 SM025
Ch. 6 Data Description

Topic: 6. DATA DESCRIPTION

Sub-Topic: 6.2 Measures of Location
6.3 Measures of Dispersion

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Learning Outcomes: At the end of this lesson, students should be able to
6.2 (a) find and interpret the mean, mode, median, quartiles and percentiles for ungrouped data.

(b) construct and interpret box-and-whisker plots for ungrouped data.

*Include: Lower fence, upper fence and outliers and data distribution interpretation.

6.3 (a) find and interpret variance and standard deviation for ungrouped data.
(c) find and interpret the Pearson’s Coefficient of Skewness.

*When coefficient is very close to 0 (negative or positive), the distribution of data is almost symmetrical.

Measures in Statistics

Measure of Location Measure of Dispersion

To determine the central value of a set of To determine the dispersion of a set of data

data (mean, median and mode). (range, interquatile range, standard deviation

and variance).

X

MEASURE OF LOCATION DEFINITON/FORMULA

x  Sum of all data or x   fx
Number of data f

 x
n
Mean Add/ subtract a constant to each score, the mean will

change by adding(subtracting) that constant.

Multiply(or divide) each score by a constant, then the
mean will change by being multiplied by that constant.

Mode The mode of a set of data is the value that occurs most
frequently.

Median/Quartile/Percentile

Interquartile Range,
IQR  Q3  Q1

Semi Interquartile Range, Pk  xxss2,xs1 , if s  ,
if s  ,
SIQR  1 Q3  Q1 
2

where s  nk and s  is the least integer greater than s.
100

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MEASURE OF DISPERSION DEFINITION/FORMULA

n 1  n xi 2 1 fixi 2
Variance     s2 i1xi2n i1 fixi2  n
Standard Deviation
n 1 or s2  n 1

s  s2  variance

Add/ subtract a constant to each score, then the
standard deviation will NOT CHANGE

Multiply(or divide) each score by a constant, the the
standard deviation will change by being multiplied by
that constant.

Box and Whisker Plots

 Another graphical representation of data.

 Construct based on the lowest value, lower quartile, Q1, median, Q2, upper quartile,
Q3  and the highest value.

 Can be represented horizontally or vertically.

Lower boundary/fence, Upper boundary/fence,

Q1  1.5Q  Q Q3  1.5Q Q
3 1 3 1

Q1  1.5IQR Q3  1.5IQR

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1.5Q3  Q1 
1.5Q3  Q1 

The Pearson’s Coefficient of Skewness

3mean  median or Sk  mean  mode
standard deviation
Sk  standard deviation

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The skewness by Pearson’s Coefficient

Skewness Skewed to the LEFT Almost Symmetrical

Pearson’s Coefficient Sk  0.1 0.1  Sk  0

Interpretation on the shape of the distribution

Skewness Skewed to the LEFT

Graphs

Measure of Location Mean  Median  Mode M
Box-Plot
Central Tendency Q2  Q1  Q3  Q2

Median

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Mathematics 2 SM025
Session 2021/2022

Symmetrical Almost Symmetrical Skewed to the RIGHT

Sk  0 0  Sk  0.1 Sk  0.1

Symmetrical Skewed to the RIGHT

Mean  Median  Mode Mode  Median  Mean

Q2  Q1  Q3  Q2 Q3  Q2  Q2  Q1

Mean Median

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EXAMPLE 5
The stem and leaf diagram shows the number of flies caught in an insect trap for 27 days.

Stem Leaf

0 112
1 23556
2 223588
3 44445779
4 26778

Key: 1 2 means 12

(a) Find
(i) mean, mode and median.

(ii) Q1, Q3 and semi interquartile range.

(iii) 81th percentile.
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(iv) variance and standard deviation.

(b) Illustrate the above data by constructing a box and whisker plot. Hence, describe the
skewness of the distribution.

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EXAMPLE 6
The table shows the distribution of grades of students for a certain subject in an examination.

Grade 123456789

Number of Students 7 13 9 7 7 2 1 1 1

(a) Find
(i) mean, mode and median.

(ii) first quartile, third quartile and P12.
(iii) standard deviation.

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(b) Construct the box and whisker plot. Hence, state the shape of distribution.

EXAMPLE 7
Given the two data as listed below:

Data I: 8, 18, 9,  10,  12,  16,  1  3,  15,  16,  13,  13

Data II: 11,  13,  13,  1  ,  2,   23,  13,  14,  15, 1  8,  20
Find the mean and standard deviation for the above data and interpret the values obtained.

Data I Data II

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EXAMPLE 8
The following is the systolic blood pressure, in mm Hg, of 10 patients in a hospital.

146 135 151 155 158 146 149 124 162 173
(a) Find the mean and mode. Describe the shape of the distribution.

(b) Find the standard deviation of the systolic blood pressure of the 10 patients. Hence, find the
Pearson’s coefficient of skewness. Comment on the distribution.

(c) Find the number of patients whose systolic blood pressures exceed one standard deviation
above or below the mean.

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Ch. 6 Data Description

Topic: 6. DATA DESCRIPTION

Sub-Topic: 6.2 Measures of Location
6.3 Measures of Dispersion

Learning Outcomes: At the end of this lesson, students should be able to
6.2 (c) find and interpret the mean, mode, median, quartiles and percentiles for grouped data.
6.3 (b) find and interpret variance and standard deviation for grouped data.

(c) find and interpret the Pearson’s coefficient of skewness.

Grouped Data

Definition

Frequency distributions A table which the values for a variables are grouped into classes.

Determined by assuming the total relative frequency is 1 or 100%.

Relative frequency Relative frequency  f

f

Class interval Bounded by the lower and upper limits of the class.

Class boundary The midpoint of the upper limit of one class and the lower limit
Class width, C of the next class.

Lower limit Upper boundary – lower boundary

The smallest value of the class limit.

Upper limit The largest value of the class limit.

Class mark/ lower limit  upper limit
Mid point, xi 2

x

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MEASURE OF LOCATION DEFINITON/FORMULA

Mean x  f1x1  f2x2  ...  fkxk
Mode f1  f2  ...  fk

k

 fixi

 i1
k
 fi
i1

xi  midpoint of the class, fi  frequency

Mode  LB  d1 d1 d2 C


LB  Lower class boundary of mode class

d1  The difference between the mode class frequency
and the previous class frequency.

d2  The difference between mode class frequency and
the class frequency after the mode class frequency.

C  Class width.

Median/Quartile/Percentile

Interquartile Range,  nk  Fk1 
IQR  Q3  Q1 100
Pk  Lk  C
fk

Semi Interquartile Range,

SIQR  1 Q3  Q1  Lk  Lower class boundary of percentile class.
2 n  Number of data or the sum of frequency.

Fk1  Cumulative frequency before percentile class.
C  Class width.

fk  Frequency of percentile class.

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MEASURE OF DISPERSION DEFINITION/FORMULA

n 1  n xi 2 1 2
i1 n i1 n
fixi
Variance     s2  xi2  fixi2 
Standard deviation
n 1 or s2  n 1

s  s2  variance

EXAMPLE 9
The following frequency table shows that the consultation time (rounded to the nearest minute)
needed by a doctor for a patient in a day.

Consultation Times, minutes Number of Patients
5
5–9 8
10 – 14 9
15 – 19 3
20 – 24 5
25 – 29

(a) Calculate the mean, mode and median.

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(b) Calculate Q1, Q3, P20 and P90.

(c) Using the answer in (a), determine the skewness of the data distribution.
(d) Find the standard deviation. Hence, find the coefficient of skewness.

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EXAMPLE 10
The following table shows the height distribution for a group of students.

Height, cm Frequency Cumulative Frequency
5 5
150  h  155 3 8
155  h  160 8 16
160  h  165 6 22
165  h  170 4 26
170  h  175 4 30
175  h  180

Find
(a) mean and median.

(b) first quartile, third quartile and interquartile range.
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(c) 10th and 70th percentile.

(d) variance and standard deviation.
(e) Pearson’s coefficient of skewness and hence, describe the distribution.

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