Dr. Romario Bhong P. Ybaez

Research Director

St. Paul University Dumaguete

Descriptive statisticsDescriptive statistics summarize the distribution of

sample scores on a single variable

Measures of Central Tendency orAverages

Measures of DispersionMeasures of LocationMeasures of LocationMeasures of SkewnessMeasures of Kurtosis

Statistical properties ofthe normal distribution

The Mean The mean of a set of numerical values is the average of the

set of values.

Note: In the explanation of the mean, the numerical values can be population values or sample values. Hence, we can can be population values or sample values. Hence, we can compute the mean for either population values or sample values.

Population vs Sample Mean The term-population mean: If the values are from an entire

population, then the mean of the values is called a population mean. It is usually denoted by (read as "mu").

Explanation of the term-sample mean: If the values are from Explanation of the term-sample mean: If the values are from a sample, then the mean of the

values is called a sample mean. It is denoted by X (read as "x-bar").

Question: Why do we use the mean as a measure of the center of a set of values?

When the deviations on the left and on the right of the sample means are added, disregarding the sign of the values, we see that when the "balancing point is the sample mean, then these sums are equal in absolute value.sums are equal in absolute value.

Thus, the mean is that central point where the sum of the negative deviations (absolute value) from the mean and the sum of the positive deviations from the mean are equal. This is why the mean is considered a measure of central tendency.

Measures ofCentral Tendency

1. Mean (X) sum of all data values divided by the number of values

Xn

Xi s are the data valuesi = 1, , n (n is the total number values)

n

XX i

i 1

Example:

The scores of 10 students in a Statisticsexam are 80, 92, 85, 87, 94, 83, 88, 91, 92,and 80. Find the mean scores of the 10students.students.

2.8710

872

10

80929188839487859080 X

When to use the mean?

1. Interval and ratio types of data2. No extreme values

3. Greatest reliability 3. Greatest reliability

Median The median of a numerical data set is the numerical value in the

middle when the data set is arranged in order.

Rules:

1. When the values in the data set is odd, the median will be the 1. When the values in the data set is odd, the median will be the middle value of the ordered array.

2. When the values in the data set is even, the median will be the average of the two middle value of the ordered array.

2. Median Examples.

a) Consider the scores of 10 students: 80, 92, 85, 87, 94, 83, 88, 91, 92, and 80, 92, 85, 87, 94, 83, 88, 91, 92, and 80.

Array: 80, 80, 83, 85, 87, 88, 91, 92, 92, 94

Md = (87 + 88)/2 = 87.5

b) The final grades of 5 high school graduating students are 90, 85, 86, 80, and 89. Find the median grade.grade.

Array: 80, 85, 86, 89, and 90.

Md = 86

Question: Why does the middle number in an ordered data set measure centraltendency? When the values from above and below the median are counted,

we see that if the "balancing point" is the sample median, then the number of values above the median balances (equals) the number of values below the median.number of values below the median.

When to use the median?

1. at least ordinal (ranked) data

2. presence of extreme values

3. open-ended distribution3. open-ended distribution

.Mode The mode of a numerical data set is the most frequently

occurring value in the data set.

Tips

1. If all the elements in the data set have the same frequency of occurrence, then the data set is said to have no mode.occurrence, then the data set is said to have no mode.

2. If the data set has one value that occurs more frequently than the rest of the values, then the data set is said to be unimodal.

3. If two elements of the data set are tied for the highest frequency of occurrence, the data set is said to be bimodal

3. Mode (Mo)

Kinds of Mode:

a. Unimodal

b. Bimodalb. Bimodal

c. Trimodal

d. Polymodal

Examples: 1.) For the scores 80, 83, 85, 87, 88, 91, 92

and 94. No mode.

2.) 80, 80, 83, 85, 87, 88, 91, 92, and 94. Mo = 80, unimodalMo = 80, unimodal

3.) For the scores 80, 80, 83, 85, 87, 88, 91, 92, 92 and 94.Mo = 80 and 92, bimodal

When to use the mode?

1. nominal or categorical data2. rough or quick estimate of a central

valuevalue

3. most popular or most typical case

Other Measures ofCentral Tendency

1. Midrange (Max + Min) / 2

Example: Consider the daily allowances of 6 high school students: P60, P50, P20, P65, P10, and school students: P60, P50, P20, P65, P10, and P15. Find the midrange of their allowances.

Array: P10, P15, P20, P50, P60, P65Midrange = (65 + 10) / 2 = 37.5

2. Weighted Mean ( Xw)

where W are weights assigned to X and k is the

k

ii

k

iii

w

W

XWX

1

1

where Wi are weights assigned to Xi and k is the number of X values being studied

Example: Suppose a high school teacher has to compute the gradeof a student in a statistics subject with the followingrequirements:

Periodic Exam 30%

Mastery Test 30%

Quizzes 20%

Assignment 15%Assignment 15%

Attendance/Project 5%

If a student gets the following score: periodic exam 89%,mastery test 88%, quizzes 85%, assignment 83%, andattendance/project 90%. What is the weighted average score ofthe student?

Xw= (89)30+88(30)+85(20)+83(15)+90(5)

100

= 87.05

[OR 89(0.30) + 88(.30) + 85(.20) +83(15) +90(0.05)]

3. Geometric mean3. Geometric mean

where Xis are positive numbers

kki XXG ...

Example: A public health officer wants todetermine the concentration of Enterococcibacteria in a certain swimming pool. Theofficer gathers the following data on differentdates: 10 Ent./100 milliliter

15 Ent. /100 milliliter1500 Ent. /100 milliliter1500 Ent. /100 milliliter

90 Ent./100 milliliter100 Ent. /100 milliliter

Find the geometric mean concentration of bacteria in that swimming pool.

Solution:

= 78.80/100 milliliter

5 1009015001510 G

= 78.80/100 milliliter4. Harmonic Mean

k

i iX

kH

1

1

Example

A car travels at a rate of 50 kph fromDgte. to Jimalalud and returns at a rate of30 kph. What is the average speed for30 kph. What is the average speed forthe trip?

Solution: kphH 5.37

301

501

2

Different Shapes ofDistributions

Source: http://faculty.vassar.edu/lowry/f0204.gif