Statistics | Mathematics Form 4

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Statistics - Mathematics Form 4

1.Making a frequency table.
A frequency table is a table showing a record of raw data arranged in preparation for analysis and interpretation to give meaningful information.
It's constructed by arranging collected data in ascending order of magnitude with their corresponding frequencies.


Example 1
The following data shows the number of goals scored by a certain team during a football premiership league matches
0,0,1,1,2,1,1,0,3,4,4,3,2,2,5,6,4,6,4,1,1,1,4,5,5.
Make a frequency table showing the distribution above.



To draw a frequency table.


Step 1:

Construct a table with three columns. The first column to read the number of goals(x). write down all the data values in ascending order of magnitude i.e.(0-6)

Step 2:

The second column to read tally, then place a tally mark at the appropriate place in the column for every data value.
When the fifth tally is reached, draw a line across the first four as shown ,

Continue this process until all data values in the list are tallied.

Step 3:

Count the number of tally marks for each data value and write it in the third column as frequency.



Solution


Grouped Data

Example 2
The following data shows the distribution of marks out of 50 in a mathematics test for 20 students.
2,7,12,11,13,14,14,4,18,19,20,15,12,23,22,32,34,31,38,37.
Makes a grouped frequency table for the above data using a class interval of 5.

Solution


Measures of central tendency: mean, median and mode.
The mean, Mode and median are values about which the distribution of a set of data is considered to be roughly balanced.
Mean
This is the sum of data values divided by the number of data values. The mean can be obtained for grouped or ungrouped data.



Example
The following are ages of college students. Calculate their mean age 27, 28, 31, 30, 32, 35, 28, 30, 29

Solution


Grouped data


Example
The table below shows the distribution of average marks out of 50 for thirty physics students in ngombini school.


Calculate the mean mark for the thirty students.



Solution
First make a frequency table as shown below.

Median
The median of a set of data values is the middle values of the set of data when it has been arranged in ascending order.

Example
37, 25, 27, 22, 28, 29, 26, 24, 25
Find the median of this set of data.

Solution
Arrange the data values in ascending order
22, 24, 25, 25, 26, 27, 28, 29, 37
Since the data values are 9, the fifth value is the middle value.

Therefore the median of the set of data is 26.
The number of the values n, in the data set = 9.

 






Mode
The mode is the most frequent occurring value in the data.

Example


Find the mode of the following set of data
38, 34, 38, 35, 32, 38, 39.

Solution

The mode is 38 since it occurs most often.


Note
It's possible for a set of data to have more than one mode.






STATISTICS

Statistics is a very vital branch of mathematics as it involves gathering of raw data, recording, analysis and interpretation of data, into meaningfull information, which helps us to make proper judgment in life.

The Assumed Mean

When the number of values in a set of data is large, the assumed mean method is used. The assumed mean is an expected value of the mean also known as the working mean, usually denoted as A. The assumed mean does not need to be one of the values given.

Estimate by calculation:
1.The median
2.The lower and upper quartiles
Solution

For one to estimate the median and the two quartiles, one must create a cumulative frequency table with lower and upper class boundary column as shown.



The Median

For us to calculate the median we must first of all identify the median class, and since we have 100 data values the middle value must lie at the 1/2 (N +1)value (refer to background) i.e. it must lie at the 1/2(100+150).5th
value in the cumulative frequency column.
Therefore the median class should be the one having 59.5-69.5 class boundaries

The median (Q2) using the formula


where
L = Lower class boundary of the median class.
N =Total frequency of the distribution
.
Cfa
= Cumulative frequency above the median class.
f
m
= Frequency of the median class.
i= Class interval of the median class
.




The lower quartile (Q1)
Identify the quartile class as you did with the median class above.
Calculate the lower quartile (Q1
) using the formula



Where
L= Lower class boundary of the Q1 class.
N= Total frequency of the distribution.
Cfa = Cumulative frequency above the Q1 class.
fQ1= Frequency of the Q1 class.
i= Class interval of the Q1 class.

In the case above:

L=39.5
N=100
Cf
a
= 24
f
Q1
= 12
i = 10



For the Upper Quartile (Q3)
Identify the upper quartile class (Q3
) and highlight it.
Calculate the upper quartile Q3
using the formula




where
L= Lower class boundary of the Q3 class.
N= Total frequency of the distribution.
Cfa = Cumulative frequency above the Q3 class.
fQ3= Frequency of the Q3 class.
i= Class interval of the Q3 class.



In the case above:

L=69.5
N=100
Cfa= 68
fQ3 = 15
i = 10


NB:The difference between the upper quartile (Q3) and the lower quartile (Q1)is called the interquartile range.
Therefore, in the case above the interquartile range is given by

Interquartile range =Q3- Q1
= 74.17 - 40.33
= 33.84

Half of the interquartile range is also known as the semi-interquartile range or the quartile deviation, i.e. semi-interquartile range (quartile deviation)


Use of Ogive/ Cumulative Frequency Curve.
An ogive/ cumulative frequency curve is obtained by plotting cumulative frequency values against the upper class boundaries.
The curve can be used to estimate the median, the lower quartile and the upper quartile.

Example
The table below represents mass of sugar in kg consumed in a college per day.



Draw an ogive (cumulative frequency curve) and use it to find:
1.The median.
2.The lower quartile3.
3.The upper quartile.
4.The interquartile range.



Solution
Construct a table showing the boundaries of the classes and the cumulative frequencies as below


Plot the cumulative frequency against the upper class boundaries to get the cumulative frequency curve shown below



.

1.The median
Q2
= (1/2 x 60)th
value. i.e. 30th value
From the graph the 30th value is 26.5

2.The lower quartile
Q1
= (1/4 x 60)th
value. i.e 15th
value.From the graph the 15th
value is 24.


3.The upper quartile

Q3
= (3/4 x 60)th
value. i.e 45th
value.From the graph the 45th
value is 28.25



4.The interquartile range = Q3 - Q1 = 28.25 - 24 = 4.25


Example 1
One hundred students were asked to estimate the distance between their dormitories and their classes in meters. Their results were shown as in the table below.


Solution
Using steps 1 to 5 in II above and taking 38 to be the assumed mean, the deviation ,d is given by x-A=d.



Example 2

The height of 50 tree seedlings in a tree nursery in cm were recorded as in the table below. Using a suitable assumed mean estimate the mean to the nearest cm.


Solution
Prepare a frequency table and obtain the midpoint(x) of each class. Follow steps 1 to 5 as previously explained.
(For grouped data we get the midpoint of each class first. The deviation are then calculated by subtracting the assumed mean A=28 from each midpoint x. ie 3-28= -25, 8-28=-20, ---, 48-28= +20)

NB: the assumed mean in both examples must be permanently highlighted
.

Making a cumulative frequency table
Cumulative frequency table is made by adding a cumulative frequency column to a frequency table.
Cumulative frequency is the gradual increase of frequencies as a result of subsequent addition.

Example.
The table below shows the distribution of masses in kg of 35 people. Make a cummulative frequency table.





Solution

Estimating the median and the quartiles
a) Calculation
Median is the value that divides a distribution of data values into two equal parts.
Quartiles (Q) are the values that divide a distribution into four equal parts. Thus the lower quartile(Q
1
) is the value below which lies the 25% (1/4) of the distribution and the upper quartile(Q3
) is the value below which lies the 75% (3/4) of the distribution.

Example
The table below shows the speed of 100 public service vehicles during a police check.



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