CHAPTER 3Data Description

OUTLINE

• 3-1 Introduction• 3-2 Measures of Central Tendency• 3-3 Measures of Variation• 3-4 Measures of Position• 3-5 Exploratory Data Analysis

OBJECTIVES

• Summarize data using the measures of central tendency, such as the mean, median, mode, and midrange.

• Describe data using the measures of variation, such as the range, variance, and standard deviation.

OBJECTIVES

• Identify the position of a data value in a data set using various measures of position, such as percentiles, deciles and quartiles.

• Use the techniques of exploratory data analysis, including stem and leaf plots, box plots, and five-number summaries to discover various aspects of data.

3-1 Introduction

• A statistic is a characteristic or measure obtained by using the data values from a sample.

• A parameter is a characteristic or measure obtained by using the data values from a specific population.

3-2 Measures of Central Tendency

Mean

Median

Mode

Mid-range

3-2 The mean (arithmetric average)

• The meanmean is defined to be the sum of the data values divided by the total number of values.

3-2 The Sample Mean

• The symbol X represents the sample mean. X is read as “X - bar”. The Greek symbol Σ is read as “sigma” and means “to sum”.

𝑋ത= 𝑋1 + 𝑋2 + ⋯+ 𝑋𝑛𝑛

= σ𝑋𝑛

The following data represent the annual chocolate sales (in millions of RM) for a sample of seven states in Malaysia. Find the mean.

RM2.0, 4.9, 6.5, 2.1, 5.1, 3.2, 16.6

million 5.77 RM =

7

16.6+ 3.2+5.1+2.1+6.5 +4.9+ 2.0 =

nX

=X

3-2 The Population Mean

• The Greek symbol µ represents the population mean. The symbol µ is read as “mu”. N is the size of the finite population.

µ = 𝑋1 + 𝑋2 + ⋯+ 𝑋𝑛𝑁

= σ𝑋𝑁

A small company consists of the owner, the manager, the salesperson, and two technicians. Their salaries are listed as RM 50,000, 20,000, 12,000, 9,000 and 9,000 respectively. Assume this is the population, find the mean.

20,000 RM = 5

9,000+9,000 +12,000 +20,000+ 50,000 =

=N

X

In a random sample of 7 ponds, the number of fishes were recorded as the following, find the mean.

23 56 45 36 28 3337

3-2 The Sample Mean for an Ungrouped Frequency Distribution

• The mean for an ungrouped frequency distribution is given by

n)Xf(

=X

f = frequency of the corresponding value X n = f

The scores of 25 students on a 4-point quiz are given in the table. Find the mean score.

Score, X Frequency, f

0 2

1 4

2 12

3 4

4 3

Score, X Frequency, f f X

0 2 0

1 4 4

2 12 24

3 4 12

4 3 12

08.225

52 ==

n

XfX

The number of balls in 17 bags were counted. Find the mean

Number of balls Frequency

5 5

6 4

7 2

8 6

3-2 The Sample Mean for a Grouped Frequency Distribution

• The mean for grouped frequency distribution is given by

Xm = class midpoint = (UCL + LCL) / 2 n

XfX

m ) (=

The lengths of 40 bean pods were showed in the table. Find the mean.

Length (cm) Frequency, f

4-8 2

9-13 4

14-18 7

19-23 14

24-28 8

29-33 5

Length (cm)

Frequency, f Midpoint, Xm f Xm

4-8 2 6 12

9-13 4 11 44

14-18 7 16 112

19-23 14 21 294

24-28 8 26 208

29-33 5 31 155

6.20 40

825=

=

825 =

1552082941124412

n

XfX

Xf

m

m

Time (in minutes) that needed by a group of students to complete a game are shown as below. Find the mean.

Time (mins) Frequency

1-3 2

4-6 4

7-9 8

10-12 5

13-15 1

ሺ𝑓∙𝑋ሻ≠ 𝑓 ∙ 𝑋

ሺ𝑓∙𝑋𝑚ሻ≠ 𝑓 ∙ 𝑋𝑚

Mean = (f•X) / n = 23 / 12 = 1.92

Mean = (f • X) / n = (12 x 10) / 12 = 10

Score, X Frequency, f f•X

0 1 0

1 3 3

2 5 10

3 2 6

4 1 4

X = 10 n = f = 12 (f•X) = 23

3-2 The Median

• When a data set is ordered, it is known as a data array.

• The median is defined to be the midpoint of the data array.

• The symbol used to denote the median is MD.

The ages of seven preschool children are

1, 3, 4, 2, 3, 5, and 1. Find the median.

1. Arrange the data in order.

2. Select the middle point.

• In the previous example, there was an odd number of values in the data set.

• In this case it is easy to select the middle number in the data array.

• When there is an even number of values in the data set, the median is obtained by taking the average of the two middle numbers.

Six customers purchased these numbers of magazines: 1, 7, 3, 2, 3, 4. Find the median.

1. Arrange the data in order.

2. Select the middle point.

3-2 The Median - Ungrouped Frequency Distribution

• For an ungrouped frequency distribution,

find the median by examining the cumulative frequencies to locate the middle value.

• If n is the sample size, compute n/2. Locate the data point where n/2 values fall below and n/2 values fall above.

LRJ Appliance recorded the number of VCRs sold per week over a one-year period.

No. Sets Sold Frequency

1 4

2 9

3 6

4 2

5 3

Total 24

• To locate the middle point, divide n by 2; 24/2 = 12.

• Locate the point where 12 values would fall below and 12 values will fall above.

• Consider the cumulative distribution.

• The 12th and 13th values fall in class 2. Hence MD = 2.

This class contains the 5th through the 13th values.

No. Sets Sold Frequency Cumulative Frequency

1 4 4

2 9 13

3 6 19

4 2 21

5 3 24

Median

3-2 The Median - Grouped

Frequency Distribution

• For grouped frequency distribution, find the median by using the formula as shown below:

Median, MD = Lm + (W)

n = sum of frequencies

cf = cumulative frequency of the class immediately preceding the median class

f = frequency of the median class

w = class width of the median class

Lm = Lower class boundary of the median class

Find the median by using the following data.

Class Frequency, f

16 - 20 3

21 - 25 5

26 - 30 4

31 - 35 3

36 - 40 2

Class Frequency, f Cumulative Frequency

16 - 20 3 3

21 - 25 5 8

26 - 30 4 12

31 - 35 3 15

36 - 40 2 17

• To locate the halfway point, divide n by 2;

17/2 = 8.5 ≈ 9.

• Find the class that contains the 9th value. This will be the median class.

• Consider the cumulative distribution.

The median class will then be 26-30.

=17 = = = –25.5 =5

( )

( ) = (17/ 2) – 8

4 = 26.125.

ncffwL

MDn cf

fw L

m

m

8430.525.5

25 25.5

( ) .

Find the median by using the following data.

Class Frequency, f

5-14 5

15-24 7

25-34 19

35-44 17

45-54 7

3-2 The Mode

• The mode is defined to be the value that occurs most often in a data set.

• A data set can have more than one mode.

• A data set is said to have no mode if all values occur with equal frequency.

Find the mode for the number of children per family for 10 selected families.

Data set: 2, 3, 5, 2, 2, 1, 6, 4, 7, 3.

Ordered set: 1, 2, 2, 2, 3, 3, 4, 5, 6, 7.

Mode: 2.

• Six strains of bacteria were tested to see how long they could remain alive outside their normal environment. The time, in minutes, is given below. Find the mode.

• Data set: 2, 3, 5, 7, 8, 10.

• There is no mode since each data value occurs equally with a frequency of one.

• Eleven different automobiles were tested at a speed of 15 mph for stopping distances. The distance, in feet, is given below. Find the mode.

• Data set: 15, 18, 18, 18, 20, 22, 24, 24, 24, 26, 26.

• There are two modes (bimodal). The values are 18 and 24.

3-2 The Mode – Ungrouped Frequency Distribution

Values Frequency, f

15 3

20 5

25 8

30 3

35 2

Mode Highest frequency

• The mode for grouped data is the modal class.

• The modal class is the class with the largest frequency.

Values Frequency, f

15.5 - 20.5 3

20.5 - 25.5 5

25.5 - 30.5 7

30.5 - 35.5 3

35.5 - 40.5 2

Modal Class

Highest frequency

3-2 The Mode – Grouped Frequency Distribution

3-2 The Midrange

The midrange is found by adding the lowest and highest values in the data set and dividing by 2.

The midrange is a rough estimate of the middle value of the data.

The symbol that is used to represent the midrange is MR.

3-2 Distribution Shapes

Frequency distributions can assume many shapes.

Three are three most important shapes:

i) Positively skewed

ii) Symmetrical

iii) Negatively skewed

Positively SkewedPositively Skewed

X

Y

Mode < Median < Mean

Positively Skewed

SymmetricalSymmetrical

n

Y

X

Symmetrical

Mean = Media = Mode

Negatively SkewedNegatively Skewed

Y

X

Negatively Skewed

Mean < Median < Mode

3-3 Measures of Variation

Measure the variation and dispersion of

the data.

Range

Variance

Standard Deviation

Coefficient of Variation

3-3 Range

The range is defined to be the highest value minus the lowest value. The symbol R is used for the range.

R = highest value – lowest value.

Extremely large or extremely small data values can drastically affect the range.

3-3 Population Variance

𝟐 = σ(𝑿− µ)𝟐𝑵

General Formula

• The symbol for population

variance is .

• X = Individual value

• µ = Population mean

• N = Population size

3-3 Population Standard Deviation

𝟐 = ඥ𝝈𝟐 = ඨσ(𝑿− µ)𝟐𝑵

General Formula

• The population standard

deviation is square root of

the population variance.

ean = (10 + 60 + 50 + 30 + 40 + 20)/6 = 210/6

= 35.

X X-µ (X-µ)2

10 -25 625

60 +25 625

50 +15 225

30 -5 25

40 +5 25

20 -15 225

∑X: 210 ∑(X-µ)2: 1750

The variance 2 = 1750/6 = 291.67.

The standard deviation = 291.67 = 17.08

A class of 6 children sat a test; the resulting marks scored out of 10, were as follows:

4 5 6 8 4 9

Calculate the mean, variance and standard deviation of this population.

3-3 Sample Variance

General Formula

• The symbol for sample

variance is .

• = Sample mean

• n = Sample size

𝑺𝟐 = σ(𝑿− 𝑿ഥ)𝟐𝒏− 𝟏

3-3 Sample Standard Deviation

𝑺𝟐 = ඥ𝑺𝟐 = ඨσ(𝑿− 𝑿ഥ)𝟐𝒏− 𝟏

General Formula

• The sample standard

deviation is square root of

the sample variance.

Mean, X = (35+45+30+35+40+25)/6 = 210/6 = 35

X X-X (X-X)2

35 0 0

45 +10 100

30 -5 25

35 0 0

40 +5 25

25 -10 100

∑X: 210 ∑(X-X)2 : 250

The variance 2 = 250/(6-1) = 50The standard deviation = 50 = 7.07

Ungrouped frequency distribution

Grouped frequency distribution

𝟐 = σ𝒇(𝑿− µ)𝟐𝑵

𝟐 = σ𝒇(𝑿𝒎 − µ)𝟐𝑵

How to find the population variance and standard deviation for data with frequency provided?

How to find the population variance and standard deviation for data with frequency provided?

• For ungrouped data, use the actual observed X value in the different classes.

• For grouped data, use the same formula with the X value replaced by class midpoints, Xm.

Ungrouped frequency distribution

Grouped frequency distribution

How to find the sample variance and standard deviation for data with frequency provided?

How to find the sample variance and standard deviation for data with frequency provided?

𝑺𝟐 = σ𝒇(𝑿− 𝑿ഥ)𝟐𝒏− 𝟏

𝑺𝟐 = σ𝒇(𝑿𝒎− 𝑿ഥ)𝟐𝒏− 𝟏

Example

X f Xm fXm (Xm - µ) (Xm - µ)2 f(Xm - µ)2

1-10 4 5.5 22 -10.59 112.1481 448.5924

11-20 8 15.5 124 -0.59 0.3481 2.7848

21-30 5 25.5 127.5 9.41 88.5481 442.7405

N = 17

∑fXm = 273.5

∑f(Xm - µ)2 =

894.1177

Mean, µ = ∑fXm / N = 273.5/17 = 16.09

Variance, 2 = 894.1177 / 17 = 52.60

Standard deviation, = 2

= 52.60 = 7.25

Question

a) The scores in a statistics test for 60 candidates are shown in the table. Find the mean, variance, and standard deviation for this population.

Score Frequency

0-19 8

20-39 13

40-59 24

60-79 11

80-99 4

b) The following table showed the number of cars passed by UM hospital in a random sample of days. Compute the sample mean, variance and standard deviation.

Number of cars Number of days

101-120 2

121-130 4

131-140 15

141-150 10

151-160 7

3-3 Coefficient of Variation

• The coefficient of variation is defined to be the standard deviation divided by the mean. The result is expressed as a percentage.

• It is used to compare the standard deviation of different units.

CVarsX

or CVar 100% 100%. =

3-3 Chebyshev’s Theorem

• 75% of the values will lie within 2 standard deviations of the mean.

• Approximately 89% will lie within 3 standard deviations.

3-3 Empirical (Normal) Rule

• For any bell shaped distribution:

Approximately 68% of the data values will fall within one standard deviation of the mean.

Approximately 95% will fall within two standard deviations of the mean.

Approximately 99.7% will fall within three standard deviations of the mean.

-- 95%

Question

The scores on a national achievement exam have a mean of 480 and standard deviation of 90. If the scores are normally distributed, find the scores for approximately 68% of the data values, 95% of the data values, and 99.7% of the data values.

3-4 Measures of Position

Measure the position of particular data in a data set.

Z - score

Percentile

Decile

Quartile

3-4 Z-Score

The standard score or z score for a value is obtained by subtracting the mean from the value and dividing the result by the standard deviation.

The symbol z is used for the z score.

Z = (value - mean)/ standard deviation

• The z score represents the number of standard deviations a data value falls above or below the mean.

For samples

zX X

sFor populations

zX

:

.

:

.

• A student scored 65 on a statistics exam that had a mean of 50 and a standard deviation of 10. Compute the z-score.

• z = (65 – 50)/10 = 1.5.

• That is, the score of 65 is 1.5 standard deviations above the mean.

• Above - since the z-score is positive.

• How about if the z-score shows a negative value?

3-4 Percentile

• Percentiles divide the distribution into 100 equal groups.

P1 P2 P3 P4 ……………… P98 P99 P100

How to find the Value Corresponding to a Given Percentile?

• Step 1: Arrange the data in order.

• Step 2: Compute c = (np)/100.

c = position value of the required percentilep = percentilen = sample size

• Step 3: If c is not a whole number, round up to the next whole number and find the corresponding value. If c is a whole number, use the value halfway between c

and c+1.

Find the value of the 25th percentile for the following data set: 18, 12, 3, 5, 15, 8, 10, 2, 6, 20.

• Is the data arranged in order?

• Arrange data in order form: 2, 3, 5, 6, 8, 10, 12, 15, 18, 20.

• n = 10, p = 25, so c = (1025)/100 = 2.5. Since 2.5 is not a whole number, round up to c = 3.

• Thus, the value of the 25th percentile is the value X = 5.

3-4 Decile

• Deciles divide the data set into 10 groups.

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10

What is the relationship between percentile and decile?

D1 corresponding to P10

D2 corresponding to P20

D9 corresponding to P90

D10 corresponding to P100

3-4 Quartile

• Quartiles divide the data set into 4 groups.

Q1 Q2 Q3 Q4

What is the relationship between percentile and quartile?

Q1 corresponding to P25

Q2 corresponding to P50

Q3 corresponding to P75

Q4 corresponding to P100

Median

3-4 Outliers and the Interquartile Range (IQR)

• An outlier is an extremely high or an extremely low data value when compared with the rest of the data values.

• The Interquartile Range, IQR = Q3 – Q1.

Procedures to identify outliers

• To determine whether a data value can be considered as an outlier:

• Step 1: Arrange the data in order.

• Step 2: Compute Q1 and Q3.

• Step 3: Find the IQR = Q3 – Q1.

• Step 4: Compute (1.5)(IQR).

• Step 5: Compute Q1 – (1.5)(IQR) and Q3 + (1.5)(IQR).

• Step 6: Compare the data value (say X) with Q1 – (1.5)(IQR) and Q3 + (1.5)(IQR).

• If X < Q1 – (1.5)(IQR) or X > Q3 + (1.5)(IQR), then X is considered an outlier.

• Find Q1.

• Q1 is corresponding to P25

c = (n•p)/100 = (8•25)/100 = 2nd • Since 2 is a whole number, take value

between 2nd and 3rd.• Q1 = (6+12)/2 = 9

• Find Q3.

• Q3 is corresponding to P75

c = (n•p)/100 = (8•75)/100 = 6th • Since 6 is a whole number, take value

between 6th and 7th.• Q3 = (18+22)/2 = 20

• Find IQR.

• IQR = 20 - 9 = 11.

• (1.5)(IQR) = (1.5)(11) = 16.5.

• 9 – 16.5 = – 7.5 and 20 + 16.5 = 36.5.

• The value of 50 is outside the range – 7.5 to 36.5, hence 50 is an outlier.

3-5 Exploratory Data Analysis - Stem and Leaf Plot

• A stem and leaf plot is a data plot that uses part of a data value as the stem and part of the data value as the leaf to form groups or classes.

Leaf

Stem

Example of Stem and Leaf Plot

4 2 5 2

5 2 5 2

Stem Leaves

Stem Leading digit

Leaf Trailing digit

At an outpatient testing center, a sample of 20 days showed the following number of cardiograms done each day: 25, 31, 20, 32, 13, 14, 43, 02, 57, 23, 36, 32, 33, 32, 44, 32, 52, 44, 51, 45. Construct a stem and leaf plot for the data.

Leading Digit (Stem) Trailing Digit (Leaf)

012345

23 40 3 51 2 2 2 2 3 63 4 4 51 2 7

3-5 Exploratory Data Analysis - Box Plot

• When the data set contains a small number of values, a box plot is used to graphically represent the data set.

• These plots involve five values:

1. Minimum value

2. Q1

3. Median

4. Q3

5. Maximum value

Example of Box Plot

Minimum value Maximum value

Q2Q1 Q3

Information Obtained from a Box Plot

• If the median is near the center of the box, the distribution is approximately symmetric.

• If the median falls to the left of the center of the box, the distribution is positively skewed.

• If the median falls to the right of the center of the box, the distribution is negatively skewed.

• If the lines are about the same length, the distribution is approximately symmetric.

• If the right line is larger than the left line, the distribution is positively skewed.

• If the left line is larger than the right line, the distribution is negatively skewed.

21 girls estimated the length of a line, in mm. The results were

51 45 31 43 97 16 18 23 34 35 35

85 62 20 22 51 57 49 22 18 27

Draw the box plot and use it to identify any outliers. Comment on the shape of the distribution of length.