TOPIC 3: STATISTICS ~ MATHEMATICS FORM 3
Calculating the Mean from a Set of Data, Frequency Distribution Tables and Histogram
Calculate the mean from a set of data, frequency distribution tables and histogram
After collecting the data, organizing it and illustrating it by means of diagrams, there is a need to calculate certain statistical measures to describe the data more precisely. There are various types of measures of central tendency – the arithmetic mean (or simply the mean), the median, and the mode. Once the measures of central tendency are found, it is easier to compare two or more sets of data.
When people are asked to find the measure of central tendency of some numbers, they usually find the total of the numbers, and then divide this total by however many numbers there are. This type of measure of central tendency is the arithmetic mean. If the n values are x1+x2+x3 ………+xn then the arithmetic mean is = x1 + x2+ ….. + xn/n
The masses of some parcels are 5kg, 8kg, 20kg and 15kg. Find the mean mass of the parcels.
John
and Mussa played for the local cricket team. In the last six batting
innings, they scored the following number of runs.John: 64, 0, 1, 2, 4,
1;Mussa: 15, 20, 13, 11 , 10, 3.Find the mean score of each player.
Which player would you rather have in your team? Give a reason.
player has the same mean score. However, observing the individual
scores suggests that they are different types of player. If you are
looking for a steady reliable player, you would probably choose Mussa.
Suppose a number a is added to or subtracted from all the data. Then a is added to or subtracted from the mean.
the n values are 𝑥!+𝑥! + 𝑥! ………+𝑥!. Multiply each by a, and
we obtain 𝑎𝑥!+𝑎𝑥! + 𝑎𝑥! ………+𝑎𝑥!. So we see that the mean
has been multiplied by a.
Exercise 1
Juma rolled a six- sided die 50 times. The scores he obtained are summarized in the following table.Calculate the mean score
| Score (x) | 1 | 2 | 3 | 4 | 5 | 6 |
| Frequency (f) | 8 | 10 | 7 | 5 | 12 | 8 |
Solution
Medium
owns a small factory. He earns about 4,000,000/- from it each year. He
employs 4 people. They earn 550,000/-, 500,000/-, 450,000/- and 400,
000/-.The mean income of these five people is(4,000,000 + 550,000 +
500,000 + 450,000 + 400,000 ÷ 5 = 1,180,000/-
If
you said to one the employees that they earned about 1,180,000/- each
year they would disagree with you. In this type of situation when one of
the values is different from the others (as in Example 2), the mean is
not the best measure of central tendency to use.Arrange the incomes in
increasing order of size as follows:
The
value that appears in the middle is called the median. In this case the
value of 500,000/- is a much better idea of the average wage earned by
the employees. The median is not affected by isolated values (sometimes
called rogue values) that are much larger or smaller than the rest of
the data.
are 50 items of data, so if you arrange them in order of size, the
positions are1 ……………….. 25 and 26 …………….. 50. The
median will be the average of the 25th and 26th number.
the table there are 8 scores of 1, followed by 10 scores of 2. This
gives you 8 + 10 = 18 numbers. These are then followed by 7 scores of 3.
This gives 18 + 7 = 25 numbers. It follows that the 25th number is a 3.
The 26th number must be the first number in the next group, which is a
4.
The Median Obtained from the Data
Interpret the median obtained from the data
- The
times of five athletes in the 100 m were: 12.5 s, 12.9s, 14.8s, 15.0s,
25.2s. Find the median time. Why is the median a better measure of
central tendency to use than the mean? - Iddi has 6 maths tests
during a school term. His marks are recorded below. Find the mean and
the median mark. Explain why the median is a better measure of central
tendency than the mean 73 78 82 0 75 86 - The table below gives
the percentage prevalence of HIV infection in female blood donors for
the years 1992 to 2003. Find the mean and median of these figures.
| 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 |
| 5.9 | 6.2 | 4.8 | 9.4 | 8.2 | 11.6 | 11.8 | 12.6 | 13.3 | 13.7 | 12.3 | 11.9 |
Explain the concept of mode
The
mode is value that occurs most often in a set of data.This is another
measure of central tendency. It is possible for data to have more than
one mode.
- If you ran a shoe shop you would want to know the most popular size.
- If you ran a restaurant you would want to know what type of food is ordered most.
- 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 5
- 58, 57, 60, 59, 50, 56, 62
- 5, 10, 10, 10, 15, 15, 20, 20, 20, 25
- 1 occurs most (3 times): The mode is 1
- All the numbers appear once: There is no mode.
- There are three 10s and three 20s: Modes are 10 and 20.
- Ten
pupils were asked how many brothers or sisters they had. The results
are recorded below. Find the mode number 0, 1 , 1, 2, 5, 0, 1 3 , 1 and
4. - Eight motorists were asked how many times they had taken the
driving test before they passed. The results are recorded below. Find
the mode number. 14213141 - Give examples of where the mode is a better measure of central tendency than either the mean or the median.
- Find the mode of these sets of numbers.
- 0, 1, 1, 3, 4, 5, 5, 5, 6, 7, 8
- 3, 8, 4, 3, 8, 4, 3, 8, 8, 3, 3, 4
- 5, 12, 6, 5, 11, 12, 5, 5, 8, 12, 7, 12
- 3, 6, 2, 8, 2, 1, 9, 12, 15
Find the mode using frequency distribution and a histogram
a set of data consists of many different values, such as heights of
people measured to the nearest centimeter. Then the data is grouped, for
example into 160 – 165 cm, and so on. If the data has been grouped
together in classes, then unless you have a list of all the individual
values, you only know approximately what each value is. For this reason,
you can only estimate the mean and the median. Also, if all the values
are different, you do not have a single value as the mode. Instead you
have a modal class, as shown in the example below.
grouped in classes can be illustrated by a histogram.Suppose one of the
intervals is from 10 to 19, where data has been rounded to the nearest
whole number. The class limits are 10 and 19. The data in this interval
could be as low as 9.5 or as high as 19.5. These are the class
boundaries. The width of the interval is the difference between the
class boundaries, in this case it is 10.
histogram consists of rectangles between the class boundaries, with
height corresponding to the frequency. The area of each rectangle is
proportional to the frequency.
Example 6
| Mark (%) | 30 – 39 | 40 -49 | 50 – 59 | 60 – 69 | 70 – 79 |
| Frequency | 5 | 3 | 20 | 2 | 10 |
- Draw a histogram
- state the modal class
For
a histogram, the horizontal axis is for the data values, and the
vertical axis is for the frequencies. So label the horizontal axis with
the marks from 30 to 80. To indicate that the axis does not start at 0
put a zig – zag to the left of 30. Label the vertical axis with
frequencies from 0 to 20. The first interval has limits 30 and 39. The
class boundaries are 29.5 and 39.5. It has a frequency of 5. So draw a
box covering the interval, and with height 5. Repeat with the other
intervals
Interpret the mode obtained from the data
Example 7
The examination results (rounded to the nearest whole number %) are given for a group of students.
| Mark (%) | 30 – 39 | 40 -49 | 50 – 59 | 60 – 69 | 70 – 79 |
| Frequency | 5 | 3 | 20 | 2 | 10 |
- Step
1: Draw a straight line from the top left hand corner of the rectangle
of the modal class, to the top left hand corner of the rectangle of the
class to the right of the modal class. - Step 2: Draw a line from
the top right hand corner of the rectangle of the modal class,to the top
right of the modal class to the left of the modal class. - Step 3: Find where these two lines intersect. This gives the mode as 54 on the horizontal axis.
- fM = frequency of the modal group
- fR = frequency of the group to the right of the modal group
- fL = frequency of the group to the left of the modal group
- W = width of the modal group
- L = lower class boundary of the modal group
