Course Title: Business StatisticsBBA (Hons)

2nd Semester

Course Instructor: Atiq ur Rehman ShahLecturer, Federal Urdu University of Arts,

Science & Technology, Islamabad+92-345-5271959

aatresh@gmail.com

Measure of Central tendency

• Mean• Median• Mode

The Mean

• It is defined as a value obtained by dividing the sum of all observations by their number.

Example

• 2 Numbers(With just 2 numbers the answer is easy: go half-way in-

between)

• what is the mean for 3 and 7?You can calculate it by adding 3 and 7 and then dividing the

result by 2:

(3+7) / 2 = 10/2 = 5

• 3 or More Numbers(You can use the same idea when you have 3 or more numbers)

Example: what is the central value of 3, 7 and 8?• You calculate it by adding 3, 7 and 8 and then dividing the results by

3 (because there are 3 numbers):

(3+7+8) / 3 = 18/3 = 6

The Median

• List all numbers in order and choose the middle one

Example

• Uncle Bob wants to know the median of age at the party, to choose an activity.

• There will be 6 kids aged 13, and also 5 babies aged 1.

• List the ages in order:1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13

• Choose the middle number:1, 1, 1, 1, 1, , 13, 13, 13, 13, 13

So, the Median age is 1313

• Sometimes there are two middle numbers. Just average them:

• What is the Median of 3, 4, 7, 9, 12, 15• There are two numbers in the middle:

3, 4, 7, 9, 12, 15• So we average them:

(7+9) / 2 = 16/2 = 8

The Median is 8

The Mode

• The Mode is the value that occurs most often

• Example: Birthday Activities (continued)• Group the numbers so we can count them:

1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13

"13" occurs 6 times, "1" occurs only 5 times, so the mode is 13

• Mode can be tricky, there can sometimes be more than one Mode.

• Example: What is the Mode of 3, 4, 4, 5, 6, 6, 74 occurs twice but 6 also occurs twice.(So both 4 and 6 are modes.)

• When there are two modes it is called "bimodal", when there are three or more modes we call it "multimodal".

Grouped Data: Arithmetic Mean

Arithmetic Mean = ΣfX/Σf

where X = Individual score f = Frequency

• From our previous example about CGPA of MCS class students,

• Arithmetic Mean = ΣfX/Σf ΣfX = 76.635Σf = 27

• Arithmetic Mean = 76.635/27 = 2.838

Artimetic Mean for IT Department MCS = 2.838

Grouped Data: MODE

• Here, l = lower limit of modal classf1 = frequency of modal classfo = frequency of class preceding the modal class.f2 = frequency of class succeeding the modal classh = size of class interval.

• Mode for IT Department:

l = 1.85, f1 = 6, fo = 0, f2 = 5, h = 0.35• Mode = 1.85+[(6-0)/(6-0)+(6-5)](0.35)

= 1.85+(6/7)(0.35) = 1.85 + 2.1/7

= 2.15• Mode for IT Department MCS = 2.15

Grouped Data: Median

• Median= L + [(n/2-Cfp)/fmed]*W• Where:

L = Lower limit of median classCfp= Cumulative frequency of class preceding the median

classFmed= Frequency of the median classW= Width of the median class (interval)N= Total number of frequenciesMedian for IT department

• Median for IT department MCS:• L = 2.57, Cfp= 11, Fmed= 3, W= 0.35, N= 27

Median=2.57+ [(27/2-11)/3)*0.35 = 2.57+ 13.5-11/3 *0.35 = 2.57+0.29166 = 2.861

Median for IT Department MCS = 2.86