LECTURE 3 - inferential statistics bmaths

LECTURE :
statisticS
OAF 112: BUSINESS MATHEMATICS AND STATISTICS
By; Jafari Selemani
selemanijafari@gmail.com I 0655 354397
1
By Jafari Selemani - 0655354397

What is statistics?
• a branch of mathematics that provides techniques
to analyze whether or not your data is significant
(meaningful)
• Statistical applications are based on probability
statements
• Nothing is “proved” with statistics
• Statistics are reported
• Statistics report the probability that similar results
would occur if you repeated the experiment

Statistics deals with numbers:
• Need to know nature of numbers collected
–Continuous variables: type of numbers associated
with measuring or weighing; any value in a
continuous interval of measurement.
• Examples:
– Weight of students, height of plants, time to flowering
–Discrete variables: type of numbers that are
counted or categorical
• Examples:
– Numbers of boys, girls, insects, plants

Populations and Samples:
• Population includes all members of a group
– Example: all 9th grade students in America
– Number of 9th grade students at SR
– No absolute number
• Sample
– Used to make inferences about large populations
– Samples are a selection of the population
– Example: 6th period Accelerated Biology
• Why the need for statistics?
– Statistics are used to describe sample populations as
estimators of the corresponding population
– Many times, finding complete information about a
population is costly and time consuming. We can use
samples to represent a population.

Measures of Central Tendency
• Find the mean
• Find the median
• Find the mode
• Make and interpret a frequency distribution
• Find the mean of grouped data
0
1
2
3
4
5
6
7
8
9

• Data set: a collection of values or measurements
that have a common characteristic.
• Statistic: a standardized, meaningful measure of a
set of data that reveals a certain feature or
characteristic of the data.
• Mean: the arithmetic average of a set of data or
sum of the values divided by the number of values.
• Median: the middle value of a data set when the
values are arranged in order of size.
• Mode: the value or values that occur most
frequently in a data set.
Key Terms

1. Find the Mean
• A business records its daily sales. These values are an
example of a data set.
Data sets can be used to:
– Observe patterns
– Interpret information
– Make predictions about future activity
 From raw (ungrouped) data, Mean is found as;
1. Find the sum of the values.
2. Divide the sum by the total number of values.
Mean = sum of values
number of values

Here’s an example:
Sales figures for the last week for the Western
Region have been as follows:
• Monday $4,200
• Tuesday $3,980
• Wednesday $2,400
• Thursday $3,100
• Friday $4,600
• What is the average daily sales figure?
• Soln:
• (4,200 + 3,980 + 2,400 + 3,100 + 4,600) ÷ 5 = 3,656

2. Find the Median
• Arrange the values in the data set from smallest to
largest (or largest to smallest) and select the value
in the middle.
• If the number of values is odd, it will be exactly in
the middle.
• If the number of values is even, identify the two
middle values. Add them together and divide by
two.

Here is an example
• A recent survey of the used car market for the
particular model John was looking for yielded
several different prices: $9,400, $11,200, $5,900,
$10,000, $4,700, $8,900, $7,800 and $9,200.
• Required: Find the median price.
• SOLN:
• Arrange from highest to lowest:
$11,200, $10,000, $9,400, $9,200, $8,900, $7,800,
$5,900 and $4,700.
• Calculate the average of the two middle values.
• (9,200 + 8,900) ÷ 2 = $9,050 or the median price

3. Find the Mode
• Find the mode in a data set by counting the number of
times each value occurs.
• Identify the value or values that occur most frequently.
• There may be more than one mode if the same value
occurs the same number of times as another value.
• If no one value appears more than once, there is no mode.
• EXAMPLE:
• Results of a placement test in mathematics included the
following scores:
65, 80, 90, 85, 95, 85, 80, 70 and 80.
• Which score occurred the most frequently?
• 80 is the mode. It appeared three times.

Make and Interpret a Frequency Distribution
• Identify appropriate intervals for the data.
• Tally the data for the intervals.
• Count the number in each interval.
• Key Terms:
• Class intervals: special categories for grouping the
values in a data set.
• Tally: a mark that is used to count data in class
intervals.
• Class frequency: the number of tallies or values in a
class interval.
• Grouped frequency distribution: a compilation of class
intervals, tallies, and class frequencies of a data set.

Example:
• Test scores on the last math test were as follows:
78 84 95 88 99 92 87 94 90 77
• REQUIRED:
• Make a relative frequency distribution using
intervals of 75-79, 80-84, 85-89, 90-94, and 95-99.

Look at this example
78 84 95 88 99 92 87 94 90 77
Class Class Relative
Interval Frequency Calculations Frequency
75-79 2 2/10 20%
80-84 1 1/10 10%
85-89 2 2/10 20%
90-94 3 3/10 30%
95-99 2 2/10 20%
Total 10 10/10 100%

4. How to Find the Mean of Grouped Data
• Make a frequency distribution.
• Find the products of the midpoint of the interval
and the frequency for each interval for all intervals.
• Find the sum of the frequencies.
• Find the sum of the products from step 2.
• Divide the sum of the products by the sum of the
frequencies.
• EXAMPLE:
• From previous data find the group mean?

Look at this example:
78 84 95 88 99 92 87 94 90 77
Product of
Class Class Midpoint and
Interval Frequency Midpoint Frequency
75-79 2 77 154
80-84 1 82 82
85-89 2 87 174
90-94 3 92 276
95-99 2 97 194
Total 10 880
Mean of the grouped data: 880 ÷ 10 = 88

Graphs and Charts:
• Interpret and draw a bar graph.
• Interpret and draw a line graph.
• Interpret and draw a circle graph.

1. Draw and Interpret a Bar Graph
• Write an appropriate title.
• Make appropriate labels for bars and scale.
• The intervals should be equally spaced and include the
smallest and largest values.
• Draw horizontal or vertical bars to represent the data.
• Bars should be of uniform width.
• Make additional notes as appropriate to aid interpretation.

Here’s an example
0 20 40 60
Product 1
Product 2
Product 3
Thousands of Units
Sales Volume
2001-2004
2004
2003
2002
2001

Interpret and Draw a Line Graph
• Make and label appropriate horizontal and vertical scales,
each with equally spaced intervals.
• Often, the horizontal scale represents time.
• Use points to locate data on the graph.
• Connect data points with line segments or a smooth curve.

Here’s an example
First Semester Sales
0
20
40
60
80
100
Jan Feb Mar Apr May Jun
Thousands
of
$
Judy Denise Linda Wally

Interpret and Draw a Circle Graph
• Find the sum of values in the data set.
• Represent each value as a fraction or decimal part of the
sum of values.
• For each fraction, find the number of degrees in the sector
of the circle to be represented by the fraction or decimal.
(100% = 360°)
• Label each sector of the circle as appropriate.

Here’s an example
43%
35%
16%
6%
Local Daycare Market Share
Teddy Bear
La La Land
Little Gems
Other

Measures of Dispersion
• Find the range.
• Find the standard deviation.
• Find the variance
• Key Terms:
• Measures of central tendency: statistical
measurements such as the mean, median or
mode that indicate how data groups toward the
center.
• Measures of variation or dispersion: statistical
measurement such as the range and standard
deviation that indicate how data is dispersed or
spread.

• Range: the difference between the highest and lowest
values in a data set. (also called the spread)
• Deviation from the mean: the difference between a value
of a data set and the mean.
• Standard variation: a statistical measurement that shows
how data is spread above and below the mean.
• Variance: a statistical measurement that is the average of
the squared deviations of data from the mean. The square
root of the variance is the standard deviation.
• Square root: the quotient of number which is the product
of that number multiplied by itself. The square root of 81 is
9. (9 x 9 = 81)
• Normal distribution: a characteristic of many data sets
that shows that data graphs into a bell-shaped curve
around the mean.

• Quartiles: Data can be divided into four regions that
cover the total range of observed values. Cut points
for these regions are known as quartiles.
• In notations, quartiles of a data is the ((n+1)/4)qth
observation of the data, where q is the desired
quartile and n is the number of observations of
data.
• An example with 15 numbers
• 3 6 7 11 13 22 30 40 44 50 52 61 68 80 94
Q1 Q2 Q3
• The first quartile is Q1=11. The second quartile is
Q2=40 (This is also the Median.) The third quartile
is Q3=61.

• Inter-quartile Range: Difference between Q3 and
Q1. Inter-quartile range of the previous example is
61- 40=21. The middle half of the ordered data lie
between 40 and 61.
• Coefficient of Variation: The standard deviation of
data divided by it’s mean. It is usually expressed in
percent. Coefficient of variation = 100

x


5. Find the Range in a Data Set
• Find the highest and lowest values.
• Find the difference between the two.
• Example: The grades on the last exam were 78, 99,
87, 84, 60, 77, 80, 88, 92, and 94.
The highest value is 99.
The lowest value is 60.
The difference or the range is 39.

Calculation of median – Grouped data:
• For calculation of median in a continuous
frequency distribution the following formula will be
employed. Algebraically,

Example: Median of a set Grouped Data in a Distribution
of Respondents by age
Age Group Frequency of
Median class(f)
Cumulative
frequencies(cf)
0-20 15 15
20-40 32 47
40-60 54 101
60-80 30 131
80-100 19 150
Total 150

Median (M)=40+
40+
=
= 40+0.52X20
= 40+10.37
= 50.37

GROUPED MODE: defined it as “the mode of a distribution
is the value at the point armed with the item tend to most
heavily concentrated. It may be regarded as the most typical of
a series of value”
The exact value of MODE can be obtained by the following
formula.
Where; L1 = Lower class limit of modal class
F1 = Frequency in modal class; F0 = frequency below modal
class; F2 = frequency above modal class; i = class interval
Z=L1+

Monthly rent (Rs) Number of Libraries (f)
500-1000 5
1000-1500 10
1500-2000 8
2000-2500 16
2500-3000 14
3000 & Above 12
Total 65
Example: Calculate Mode for the distribution of monthly
rent Paid by Libraries in Karnataka

Z=2000+
Z =2000+
Z=2400
Z=2000+0.8 ×500=400

Find the Standard Deviation
• The deviation from the mean of a data value is the
difference between the value and the mean.
• Get a clearer picture of the data set by examining
how much each data point differs or deviates from
the mean.
• When the value is smaller than the mean, the
difference is represented by a negative number
indicating it is below or less than the mean.
• Conversely, if the value is greater than the mean,
the difference is represented by a positive number
indicating it is above or greater than the mean.

• Find the mean of a set of data.
• Mean = Sum of data values
Number of values
• Find the amount that each data value deviates or is
different from the mean.
• Deviation from the mean = Data value - Mean
• Here’s an example:
• From the following data set, find the deviation from
the mean;
• Data set: 38, 43, 45, 44

• Mean = 42.5
• 1st value: 38 – 42.5 = -4.5 below the mean
• 2nd value: 43 – 42.5 = 0.5 above the mean
• 3rd value: 45 – 42.5 = 2.5 above the mean
• 4th value: 44 – 42.5 = 1.5 above the mean
• Interpret the information:
• One value is below the mean and its deviation is
-4.5.
• Three values are above the mean and the sum of those
deviations is 4.5.
• The sum of all deviations from the mean is zero. This is true
of all data sets.
• We have not gained any statistical insight or new information
by analyzing the sum of the deviations from the mean.

Find the standard deviation of a set of data
• A statistical measure called the standard deviation uses
the square of each deviation from the mean.
• The square of a negative value is always positive.
• The squared deviations are averaged (mean) and the
result is called the variance.
• The square root is taken of the variance so that the
result can be interpreted within the context of the
problem.
• This formula averages the values by dividing by one less
than the number of values (n-1).
• Several calculations are necessary and are best
organized in a table.

Steps in finding SD:
1. Find the mean.
2. Find the deviation of each value from the mean.
3. Square each deviation.
4. Find the sum of the squared deviations.
5. Divide the sum of the squared deviations by one
less than the number of values in the data set.
This amount is called the variance.
6. Find the standard deviation by taking the square
root of the variance.

EXAMPLE:
Find the standard deviation for the following data
set: 18 22 29 27
Deviation Squares of
Value Mean from Mean Deviation
18 24 18 – 24 = -6 -6 x -6 = 36
22 24 22 – 24 = -2 -2 x -2 = 4
29 24 29 – 24 = 5 5 x 5 = 25
27 24 27 – 24 = 3 3 x 3 = 9
Sum of Squared Deviations 74

Variance = sum of squared deviations
n – 1
Variance = 74 ÷ 3 = 24.666667
Standard deviation = square root of the variance
Standard deviation = 4.97 rounded

• A large variance means that the individual scores
(data) of the sample deviate a lot from the mean.
• A small variance indicates the scores (data) deviate
little from the mean
• Variance helps to characterize the data concerning a
sample by indicating the degree to which individual
members within the sample vary from the mean.

Probability Distributions:
• Inferential statistical methods use sample data to make
predictions about the values of useful summary
descriptions, called parameters, of the population of
interest. This part treats parameters as known
numbers.
• We first define the term probability, using a relative
frequency approach.
• The probability distribution of the random variable X
lists the possible outcomes together with their
probabilities the variable X can have.
• From the probability distribution, The mean and the
standard deviation of the discrete random variable are
defined in the following ways.

• The variance and standard deviation will be;

LECTURE 3 - inferential statistics bmaths

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