Write your own
error: Content is protected !!
MATHEMATICS: FORM THREE: Topic 3 - STATISTICS
TOPIC 3: STATISTICS
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
The masses of some parcels are 5kg, 8kg, 20kg and 15kg. Find the mean mass of the parcels.
Total mass = (5 + 8 + 20 + 15) kg = 48kg
The number of parcels = 4
The mean mass = 48 kg ÷ 4 = 12 kg
The arithmetic mean used as measure of central tendency can be misleading as can be seen in the following example.
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.
John’s mean = (64 + 0 + 1 + 2 + 4 + 1) ÷ 6 = 12
Mussa’s mean = (15 + 20 + 13 + 11 + 10 + 3) ÷ 6 = 12
Each 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.
Often it is possible to use the mean of one set of numbers to find the mean of another set of related numbers.
Suppose a number a is added to or subtracted from all the data. Then a is added to or subtracted from the mean.
Suppose 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.
Interpreting the Mean Obtained from a Set Data, Frequency Distribution Tables and Histogram
Interpret the mean obtained from a set data, frequency distribution tables and histogram
The Concept of Median
Explain the concept of median
Mr.Samwel 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.
If the data consists of an even number of values, find the mean of two middle values as shown in the next example.
The Medium from a Set of Data
Calculate the medium from a set of data
Find the median of the numbers: 12, 23, 10, 8, 22, 14, 30, and 18.
Arranging in increasing order of size, we get 8 10 12 14 18 22 23 30
Median = (14 + 18) ÷ = 16
The Median using Frequency Distribution Tables and Cumulative Curve
Find the median using frequency distribution tables and cumulative curve
Juma rolled a six- sided die 50 times. The scores he obtained are summarized in the following table. Calculate the modianl score
here 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.
In 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 is then = (3 + 4) ÷ 2 = 3.5
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.
The Concept of Mode
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.
Data with two modes are said to be bi – modal. Why mode? The mode is often important to know. For example:
- 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.
Calculate the mode
State the mode for the following sets of numbers:
- 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
Finding the Mode using Frequency Distribution and a Histogram
Find the mode using frequency distribution and a histogram
Interpreting the Mode Obtained from the Data
Interpret the mode obtained from the data
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|
To estimate the mode, there are two methods.
By drawing:Use the histogram of the first part.Then proceed as follow;
- 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.
By calculation: Let
- 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