Recap

Review the central tendencies – Mean, Median, Mode

Understand the different types of statistical representations

Central Tendencies - 3 M’s

Mean - the arithmetic mean, is commonly called the average.

Median - the middle data/scorethe middle of a distribution: half the scores are above the median and half are below the median.

Mode is the observation that occurs most frequently in the sample.

x

Example (Mean, Median & Mode)Consider a XXX company,

The CEO makes $100,000 per year,Two managers make $50,000 per year,Fifteen factory workers make $15,000 eachTwo trainees make $9,000 per year.

Mean = $22,150 But only three out of 20 persons having salary greater than $22K!Median = $15,000Mode = $15,000What about Measure of Spread?

Another Example A professor of statistics wants to report the results of a midterm

exam, taken by 100 students.

Marks

Mean 73.98Standard Error 2.1502163Median 81Mode 84Standard Deviation 21.502163Sample Variance 462.34303Kurtosis 0.3936606Skewness -1.073098Range 89Minimum 11Maximum 100Sum 7398Count 100

Marks

Mean 73.98Standard Error 2.1502163Median 81Mode 84Standard Deviation 21.502163Sample Variance 462.34303Kurtosis 0.3936606Skewness -1.073098Range 89Minimum 11Maximum 100Sum 7398Count 100

The mean provides informationabout the over-all performance level of the class. It can serve as a tool for making comparisons with other classes and/or other exams.

The mode must be used when data is qualitative. If marks are classified by letter grade, the frequency of each grade can be calculated.Then, the mode becomes a logical measure to compute.

Example (cont’)

Marks

Mean 73.98Standard Error 2.1502163Median 81Mode 84Standard Deviation 21.502163Sample Variance 462.34303Kurtosis 0.3936606Skewness -1.073098Range 89Minimum 11Maximum 100Sum 7398Count 100

Marks

Mean 73.98Standard Error 2.1502163Median 81Mode 84Standard Deviation 21.502163Sample Variance 462.34303Kurtosis 0.3936606Skewness -1.073098Range 89Minimum 11Maximum 100Sum 7398Count 100

The Median indicates that half of the class received a grade below 81%, and half of the class received a grade above 81%.

Interesting Facts about 3 M’s

Looking at their relationship

If a distribution is symmetrical, the mean, median and mode coincide

Interesting Facts about 3 M’s

If a distribution is non symmetrical, and skewed to the left or to the right, the three measures differ.

A positively skewed distribution(“skewed to the right”)

MeanMedian

Mode MeanMedian

Mode

A negatively skewed distribution(“skewed to the left”)

Interesting Facts about 3 M’s

Mean is called economic indicator. It can provide sample total. Median is called social indicator. It can better identify a typical value.Mean of a subgroup can be integrated to get the total mean, but not median.Trimmed mean is an alternative mean that is resistant to outliers. 

Example 1

Given a set of 11 numbers: 3, 5, 5, 5, 6, 7, 7, 8, 8, 9, and 14, find the mean, median and mode of this set of data.

Mean: 7

Median: 7

Mode: 5

How to describe the data?

Skewed to the Right / Left ?

Any possible outliers?

Graphical Presentation of Qualitative Data

Pie Chart

Bar Chart

Line Chart

Graphical Presentation of Quantitative Data

Dot Diagram• useful in displaying a small body of data, say up to

about 20 observations.

• example : height of students in a class.

140cm 150cm 160cm 170cm 180cm

Another Example (Changi International Airport)

A study at the Changi airport regarding the flight delay times for 19 flights are shown as below:

-8 -6 180 10 5 -15 20 -49 28 12

32 -23 29 -19 2 35 13 11 9

0 100 200-100-200

Graphical Presentation of Quantitative DataStem-and-leaf Diagram

• a good way to obtain informative visual display of data.• we divide each number into two part: a stem, consisting of

one or more leading digit and leaf, consisting of remaining digits.

• Example (height)

Stem LeafFrequency

14 8 1

15 2 5 6 2 4

16 1 2 3 7 8 6 9 7

17 1 2 7 5 6 5

18 2 1

Variance and standard Deviation

Measure of Spread

Box-Plot

Understand and Use Variance and Standard Deviation

Measure of SpreadRange

the difference between the largest and the smallest values Highest Value - Lowest Value

Interquartile Rangethe difference between the 75th percentile (often called Q3) and the 25th percentile (Q1).Q3-Q1

Standard Deviation, the square root of the variance.

So, what is variance, ?

a measure of how spread out a distribution is

It is computed as the average squared deviation of each number from its mean.

For example, for the numbers 1, 2, and 3, the mean is 2 and the variance is:

2(1 2)2(2 2)2(3 2)23

0.667

2

Box Plot

3 4 5 6 7 8 9 10 11 12 13 14

Box Plot

is sometimes also called the Five Number Summary.

Using Box Plots

Bigger the spread, bigger the magnitude of the range, interquartile range, and standard deviation

Low High

Example 2

A group of boys (HCI-2Mers) score an average of 75% in a test while the girls (NYG-2Modest) obtained an average of 70%. On the other hand, the standard deviation of the scores for the boys is 10% while that of the girls is only 5%. In your opinion, which group has done better? Why?

Some Important Information

Boys, SD 10%

Mean = 75%

Girls, SD5%

Mean = 70%

68% of

Boys/Girls65% - 85% 65% - 75%

95% of

Boys/Girls55% - 95% 60% - 80%

99.7% of

Boys/Girls45% - 100% 55% - 85%

2

3

Computing Variance

Step 1: Obtain sum of squares

Step 2: Obtain mean of data

Step 3: Use

x2x

22 1

x xn

Example 3

Find the mean, variance and the standard deviation for the set of data {24, 27, 27, 30, 31, 31, 31, 35, 36, 38}

Mean: 31

Variance: 18.2

Standard deviation: 4.266

Example 4

200 people were surveyed to indicate the number of days, X days, they had dinner at home with their families last week. The following shows the frequency table for the data collected.

•No. of days

•0 •1 •2 •3 •4 •5 •6 •7

•Frequency

•13 •25 •30 •55 •42 •20 •8 •7

Example 4

Find the mean, variance and standard deviation of X.

Mean: 3.075

Variance: 2.76

Standard deviation: 1.66

Additonal Information

Example 3

Find the mean, variance and the standard deviation for the set of data {24, 27, 27, 30, 31, 31, 31, 35, 36, 38}

Mean: 31

Variance: 18.2

Standard deviation: 4.266

Computing Variance

Step 1: Obtain sum of squares

Step 2: Obtain mean of data

Step 3: Use

x2x

22 1

x xn

Example 4

Find the mean, variance and standard deviation of X.

Mean: 3.075

Variance: 2.76

Standard deviation: 1.66