Mean Calculating the Mean from a Set of Data, Frequency Distribution Tables and Histogram Caleulate the mean from a set of data frequency distribution tables and histogram Measures of central tendency: calculate certain statistical measures to describe the data more precisely. ‘There are various types of data The arithmetic mean 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, Ifthe n values are x1+2+3

= Xp +X gt nt Xn

n Eeaaipled The masses of some parcels are Skg, 8kg, 20kg and 15kg, Find the mean mass ofthe parcels

Solution

‘The arithmetic mean used as measure of central tendency can be misleading as can be seen in the following example

Example2

John and Mussa played for the local cricket team, In the last six batting innings, they scored the following number of runs John: 64,00, 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

Solution

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!+2! +2! …..+x!. Multiply each by a, and we obtain az!+ax! + ax!

+axt, 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 disiribuiton tables and histogram Measures of central tendency from frequency tables os table.Calculate the mean score Medium

The Concept of Median

MrSamwel owns a small factory. He cams about 4,000,000/- from it cach year. He employs 4

in increasing order of size as follows: i 400,000 450,000 500,000 550,000 4,000,000 Median The value that appears inthe middle is called the median. In this case the value of $00,000/-is a much better idea of the average wage cared by the employees. The median is not affected by isotated values (sometimes called rogue values) that are much larger or smaller than the rest of the data 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 Caleulate the medium from a set of data

Example3

Find the median of the numbers: 12, 23, 10, 8, 22, 14,30, and 18.

Solution

Arranging in increasing order of size, we get 8 10 12 14 18 22 23 30

‘The Median using Frequency Distribution Tables and Cumulative Curve Find the median using frequency distribution tables and cumulative curve

Example 4

Juma rolled a six- sided die $0 times. The scores he obtained are summarized in the following table Calculate the modianl score

Solution

here are 50 items of data, so if you arrange them in order of size, the positions are! 25 and 26 50. he median will be the average of the 25th and 26th number.

that the 25th number isa 3. The 26th number must be the first rumber in the next group, which is a4

‘The Median Obtained from the Data Interpret the median obtained from the data 1 The times of five athletes in the 100m were: 12.5 5, 12.9, 14.85, 15.0s, 252s, Find the modian time, Why is the median @ better measure of central tendency to use than the mean?

• di 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 2 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. 19921993 1994 1995 19961997 1998 199 2000 2001 20022008 S962 48 94 82 16 118 126 33 B73 Le Mode ‘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: a. If you an a shoe shop you would want to know the most popular size a pennant

The Mode

Example

‘State the mode for the following sets of numbers: os eS errs ea e eae ie © There are three 10s and three 20s: Modes are 10 and 20. a rer nando yeeianti caine passed. The results are recorded below. Find the mode number. 14213141 ee Be sa eee rere cane © 5,12, 6, 5, 11, 12, $, 5,8, 12,7,12 d 3,6, 2,8.2,1,9, 12,15 Finding the Mode using Frequency Distribution and a Histogram.

Grouped data Suppose a set of data consists of many different values, such as heights of people measured to data has been grouped together in classe, then unless you havea ist of al the individual values, mean and the median. Also, if all the values are different, you do not have a single value as the Data grouped in classes can be illustrated by a histogram Suppose one of the intervals i fom 10 The data in this interval could be as low a8 9.5 or as high as 19-5. These are the class boundaries

The histogram consists of rectangles between the class boundaries, with height corresponding to the Hequeney. The area ofeach rectangle is proportional to the frequency.

Example 6

Mark) 30-9 0» 50-39 60-69 0-9 Freeney 5 3 2» 2 w a Daw a histogram b. state the moda class

Solution

frequencies. So label the horizontal axis with the marks from 30 to 80. To indicate that the axis does not start at puta zigzag to the left of 30. Label the vertical axis with frequencies ftom 0 20 oid co coo eg

EEE EEE EEF EE

i 1 1 Iw L —t e m Interpreting the Mode Obtained from the Data as students. i

» T T

i CI

Frequency I Cl

vo ia LV a a a a) Mask By drawing:Use the histogram of the first part Then proceed as follow;

• Step 1: Draw a straight line from the top left hand comer of the rectangle of the modal

‘lass, to the top left hand comer of the rectangle of the class to the right of the modal class,

• Step 2: Draw a line from the top right hand comer 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 $4 on the horizontal

axis By caleulation:Lct

• M=frequeney of the modal group
• R= 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

n The mode = L + —i#—fW— Qfm-fa-f)

• (20-3)x10

• The mode is $4.4

n