Measure of Dispersion Part (2)

Standard Deviation

Deviation: the difference between an individual value in a set of data and the mean for the data.

Standard Deviation is the square root of the variance. 

        

Key for formula:

FORMULA 

Standard Deviation for Grouped Data

Extra Key:

f_i = the frequency for a given interval

m_i = the midpoint of the interval

FORMULA

Example 1 (Ungrouped - Population):

The table below shows the height of 5 friends:

NAME	HEIGHT (m)
Barry	1.60
Michael	1.70
David	1.90
Steven	1.65
Ronaldo	1.85

Find out the Mean, the Variance, and the Standard Deviation.

Find the mean

Find the variance:

1^st step: find the difference of each height and the average (mean):

Barry = 1.60 – 1.74

            = -0.14m

Michael = 1.70 – 1.74

               = -0.04m

David = 1.90 – 1.74

            = -0.16m

Steven = 1.65 – 1.74

              = -0.09m

Ronaldo = 1.85 – 1.74

                = -0.11m

2^nd step: Take each difference, square it, and then average the result:

Standard Deviation (Square root of Variance)

Measure of Dispersion part (1)

The measures of dispersion are used examine the spread of the data. These are the methods used:

• Range
• Quartiles
• Variance
• Standard Deviation

RANGE

Range is the simplest method for measuring dispersion. It is the difference between the largest value and the smallest value in the data set. 

R = X_max – X_min

Example:
3, 5, 7, 9, 16, 18, 25

Range, R = 25 – 3

   = 22

QUARTILE

Quartiles divide a set of ordered data into four groups with equal numbers of values.

The interquartile range (IQR) is Q₃ – Q_1.

Example:

Calculate the median and interquartile of Brunei Darussalam’s temperature during the day in the following dates.

Statistical Representation part (2)

STEM AND LEAF

A Stem and Leaf Plot is a special table where each data value is split into a ‘stem’ (the first digit or digits) and a ‘leaf’ (usually the last digit). The stem values are listed down, and the leaf values go right (or left) from the stem values.

Example 1:

Create a stem and leaf plot for this data:

96, 86, 61, 91, 75, 84, 94, 90, 70, 92, 100, 94

Answer:

First, put this data in order

61, 70, 75, 84, 86, 90, 91, 92, 94, 94, 96, 100

We will use 6, 7, 8, 9 and 10 as stems. The plot is displayed below:

Example 2:

This is the following numbers of ages people who attended FM’s gym. Use a Stem-and-Leaf Plot to Find Mean, Median, Mode and Range.

20, 40, 33, 25, 17, 26, 32, 23, 15, 37, 43, 20, 45, 29, 48, 29

Answer:

15, 17, 20, 20, 23, 25, 26, 29, 29, 32, 33, 37, 40, 43, 45, 48

We will use 6, 7, 8, 9 and 10 as stems. The plot is displayed below:

Mean:

The mean is the average of a set of data.

= 482

Median:

The median is the middle number of a set of data.

15, 17, 20, 20, 23, 25, 26, 29, 29, 32, 33, 37, 40, 43, 45, 48

=29

Mode:

The mode is the number that occurs the most in a set of data.

15, 17, 20, 20, 23, 25, 26, 29, 29, 32, 33, 37, 40, 43, 45, 48

=20, 29

Range:

The range is the difference between the smallest value and the greatest value.

15, 17, 20, 20, 23, 25, 26, 29, 29, 32, 33, 37, 40, 43, 45, 48

48-15

=33

Statistical Representation part (1)

Summarizing large amounts of data in a compact format that yields meaningful information concerning the data could be a problem. With statistical representation, it is possible to present the data concisely without displaying the values for each observation taken.

Statistical data can be represented in various ways, depending on the given data and information. These are the common types of statistical representation used:

• Line Chart
• Bar Chart
• Pie Chart
• Pictograms

LINE CHART

A line graph is also called a line chart. It is a graphical display of information that changes continuously over time. The connected points either be increases or decreases.

The graph must have a title, both x- and y-axis, and a key.

Example:

Questions:

a)   Which day had the least amount of push-ups completed? 

Answer : Friday with a number of zero push-ups.

b)  Which day had the most amount of push-ups completed?

 Answer: Monday & Tuesday (25 push-up)

c)  Which day had a significant change in the amount of push-ups? 

Answer:  Wednesday (a decrease of 20 push-ups) and Saturday (an increase of 20 push-ups). 

BAR CHART

A Bar Graph also known as Bar Chart is a graphical display of data using bars of different heights. It can be horizontal and vertical graphs. One axis of the chart shows the specific categories being compared, and the other axis represents a discrete value.

Example 1: Vertical Bar Graph

In May 2016, this is the grade for Diploma IT students.

Grade	A	B	C	D
Students	11	3	5	4

Example 2: Horizontal Bar Graph

A gardener survey which kind of fruit for customers and families liked the most:

Fruits	Apple	Grape	Banana	Orange	Watermelon	Melon
Customer	34	20	23	12	25	9
Families	12	8	11	5	7	10

PIE CHART

A pie chart or known as circle chart is a circular statistical graphic and special chart that uses ‘pie slices’ to show relative sizes of data.

Example 1:

The pie chart below shows the grade achieved by the 23 students.

Grade	A	B	C	D
Students	11	3	5	4
Angle in Pie Chart (o)	172	47	78	63

Example 2:

The pie chart below shows the percentages of football clubs supported by 350 who loved football in a school.

Manchester United Fc: 40%

Liverpool Fc: 30%

Arsenal Fc: 15%

Chelsea Fc: 10%

Manchester City Fc: 5%

Question:

a)  How many students, in the school supports Manchester United Fc?

b)  How many students do not like Chelsea Fc?

c)  How many student favor Manchester United Fc over Liverpool Fc?

PICTOGRAMS

Pictograms are charts in which icons represent numbers to make it more interesting and easier to understand. A key is often included to indicate what each icon represents. All icons must be of the same size, but a fraction of an icon can be used to show the respective fraction of that amount.

Example:

Day	Box of food sent
Monday	10
Tuesday	51
Wednesday	19
Thursday	27
Friday	8

Can be graphed as below:

As the values are rounded to the nearest 5 boxes, the third icon on Thursday is the left half of the original.