Central Tendency
Central Tendency - Computing and understanding averages Introduction of Central Tendency
Measures of central tendency describe the average or are representative of an entire group of values or a set of data. They are most widely used, not only in academic research, but also for quantitative information reported to the public. In this lesson, we will study the three commonly used central tendency measures: mean, median, and mode. These central tendency measures are different in conception and calculation. They represent different notions of the center of a distribution. In normal distribution, mean (X ), median (Md), and mode (Mo) are equal (Figure A). In skewed distribution, mean, median, and mode can differ greatly (Figures B & C).
Computing and Understanding Mean
Mean is defined as the average value of the distribution and is also called the typical, average, or the most central value in the sample or group. The computation is simply the sum of The objectives of this lesson are:
• Understanding measures of central tendency
• Computing the mean of a set of scores by hand
• Computing the median fir a set of scores by hand
• Determining/calculating the mode of a set of scores by hand
• Selecting a measure of central tendency
• Computing mean, median, and mode using SPSS window software
X Md Mo XMd Mo Mo Md X A B C X = Md = Mo
Normal Negatively Skewed Positively Skewed
all the values in a sample/group, divided by the number of values in that sample/group. The common symbol for the mean of a sample/group of values is X (X bar) or the letter M. The mean is sensitive to extreme scores. Extreme values cause the mean to be less representative of the set of values, and therefore, less useful as a measure of central tendency.
The equation for computing a simple mean is shown in Equation 1-1
X = 1-1 Where:
X (X bar) is the average value of the sample/group of values or simply called mean.
∑, the Greek letter sigma, is the summation sign, which means add together whatever follow it Xi represents all values; X1, X2, …., Xi; in the sample/group. i = number of group
n is the size of the sample from which you are computing the mean Computing a Simple Mean by Hands
<Example 1> Computing a simple mean
If you want to know the average age of people with spinal cord injury who work at the Angel Photoshop, you would compute the mean of their ages. Here is a list of the data (Table 1): Table 1
Steps for computing a simple mean by hand: 1. List the entire set of scores/values. D Xi
2. Sum all the scores/values D ∑Xi= X1+X2+…+Xn
3. Identify the number of values in the sample/group D n 4. Divide the total or sum by the number of values D ∑Xi
n ∑Xi
25 + 27 + 32 + 28 X1+X2+X3+X4
n 4
112
4
Name Age (years old)
Mary 25 (X1) David 27 (X2) Mark 32 (X3) Jane 28 (X4) X = = = = = 28 Let’s follow the steps listed above to calculate the mean age of these four people. Step 1: List the entire set of values, which is the ages of the four people.
25, 27, 32, 38
Step 2: Add all the values (ages), which is the total of the four people’s ages.
∑Xi= X1+X2+ X3+X4 = 25 + 27 + 32 + 38 = 122
Step 3: Identify number of value in n, which is the number of people. n = 4
Step 4: Divide the total or sum by the number of values, dividing the sum of all four people’s ages by the number of people (divide step 2 by step 3)
∑Xi n = 4
122
= 30.5
Now, we can state, according to the computation above, that 30.5 is the average age of the sample of four people with spinal cord injuries who work at the Angel Photoshop.
<Example 2> Computing a weighted mean for one sample/group
Extending the above example, you can apply the equation for calculating a simple mean if you have more than one person in the same age group; however, it is easier to compute a weighted mean using Equation 1-2. A weighted mean is calculated by multiplying the value by the frequency of its occurrence.
∑Xi
X•= ∑ • njXi n = n X n X njXi n n nj 1 1 2 2 1 2 + + + + + + .... ... 1-2 Where:
X• is the weighted average value of the sample/group of values or simply the weighted mean.
∑ is the Greek letter sigma, is the summation sign, which means add together whatever follows it.
Xi represents all values; X1, X2, …., Xn; in the sample/group.
nj represents the number of subjects (frequency) in each value, n1 (the number of subjects,
frequency, in value 1, X1), n2 (the number of subjects, frequency, in value 2, X2), …, and nj (the
number of subjects, frequency, in value i, Xi). n• is the total number of subjects in the sample. Computing a Weighted Mean by Hand
<Example 3> Computing a weighted mean for more than one sample/group Sample data: Ages of a group of people:
25, 28, 32, 27, 25, 25, 32, 28, 28, 32, 27, 32, 27, 27, 32 Steps for computing a weighted mean by hand:
1. Create a table and tally the frequency for each value D n1, n2, …, nj
2. Compute the total number of subjects by summing the numbers from n1 tonj D
n•= n1 + n2 + …+ nj
3. Multiply the number of each value (Xi) to its frequency D ∑njXi = (X1*
n1)+(X2* n2)+…+(Xn* nj) Note: “*” means multiply
4. Divide the product of the values and frequency, ∑njXi, in Step 2, by the total number of values, n• D ∑
•
njXi n
Step 1: Create a table. Type the values (Xi, age) in column one and tally the frequency for each value/age in column two. In other words, how many people are age 25, how many people are age 27, and etc.
Table 2
Step 2: Compute the total number of subjects by adding the numbers from n1 tonj (in this case,
nj= n4, because we have four values/ages in the table). D n•= n1 + n2 + …+ nj
n•= n1 + n2 + n2+ n4= 3+4+3+5= 15
Step 3: Multiply the number of each value (Xi) to its frequency (nj). D ∑njXi = (X1* n1)+(X2*
n2)+…+(Xi* nj)
njXi
∑ = (X1* n1)+(X2* n2)+ (X3* n3)+(X4* n4)
= (25*3)+(27*4)+(28*3)+(32*5)=75+108+84+160=427 Age Frequency Age * Frequency
25 (X1) 3 (n1) 27 (X2) 4 (n2) 28 (X3) 3 (n3) 32 (X4) 5 (n4)
Total
Age Frequency Age x Frequency 25 (X1) 3 (n1)
27 (X2) 4 (n2)
28 (X3) 3 (n3)
32 (X4) 5 (n4)
Step 4: Divide the product of the values and frequency, ∑njXi, in Step 2 by the total number of values, n• D ∑ • njXi n X•=∑ • njXi n = 427 15 ≅28 5.
You may compute the weighted mean by plugging the numbers into Equation 1-2:
X• =∑ • njXi n = n X n X n X n X n n n n 1 1 2 2 3 3 4 4 1 2 3 4 + + + + + + = 3 4 3 5 ) 32 * 5 ( ) 28 * 3 ( ) 27 * 4 ( ) 25 * 3 ( + + + + + + =75 108 84 160 15 + + + =427 15 ≅28 5. (28.488)
According to the result of the above computation, we conclude that the weighted mean is approximately (≅) 28.5 years old, which represents the average age of the fifteen people in the sample/group.
Computing a weighted mean for more than one sample/group when ns are EQUAL and
UNEQUAL
A similar equation is applied when calculating the mean. First, calculate the mean for each sample/group using equation 1-1, then apply Equation 1-3-1 when the number of people in each group (nj) are equal, otherwise, apply Equation 1-3-2.
Age Frequency Age x Frequency
25 (X1) 3 (n1) 75
27 (X2) 4 (n2) 108
28 (X3) 3 (n3) 84
32 (X4) 5 (n4) 160
<Example 4> Computing a weighted mean for more than one sample/group when ns are EQUAL X• = n X j n ∑ • = n X X X j n j ( 1+ 2+ +.... ) ∗ 1-3-1 Group
(j=3) Number of People in the Group (nj) Group Mean (Xi)
Blue (1) 5 (n1) 12 (X1) Red (2) 5 (n2) 16 (X2) Yellow (3) 5 (n3) 14 (X3) EQUAL ns: n1 = n2 = n3 = 5 X• = n X j n ∑ • = n X X X n j ( 1+ 2+ 3) ∗ = 5 12 16 14 5 3 ( + + ) ∗ = 5 42 15 ∗ = 210 15 = 14
<Example 5> Computing a weight mean for more than one sample/group when ns are UNEQUAL X• = ∑ • nj X j n = n X n X nj X j n n nj 1 1 2 2 1 2 + + + + + + .... ... 1-3-2 Group
( j=3) Number of People in the Group (nj) Group Mean (Xi)
Blue (1) 5 (n1) 12 (X1) Red (2) 9 (n2) 16 (X2) Yellow (3) 7 (n3) 14 (X3) X• = ∑ • nj X j n = n X n X n X n n n 1 1 2 2 3 3 1 2 3 + + + + + = 5 12 9 16 7 14 5 9 7 ∗ + ∗ + ∗ + + = 60 144 84 20 + + =288 20 =14 4.
