SlideShare uma empresa Scribd logo
1 de 9
2013/05/22
1
STATISTICS
X-Kit Textbook
Chapter 9
Precalculus Textbook
Appendix B: Concepts in Statistics
Par B.2
CONTENT
THE GOAL
Look at ways of summarising a large
amount of sample data in just one or two
key numbers.
Two important aspects of a set of data:
•The LOCATION
•The SPREAD
MEASURES OF CENTRAL TENDENCY
(LOCATION)
Arithmetic Mean (Average)
Mode (the highest point/frequency)
Median (the middle observation)
Number of fraudulent cheques received at a
bank each week for 30 weeks
Week
1
2 3 4 5 6 7 8 9 10
5 3 8 3 3 1 10 4 6 8
Week
11
12 13 14 15 16 17 18 19 20
3 5 4 7 6 6 9 3 4 5
Week
21
22 23 24 25 26 27 28 29 30
7 9 4 5 8 6 4 4 10 4
ARITHMETIC MEAN
• 𝒙 =
𝟏𝟔𝟒
𝟑𝟎
= 𝟓. 𝟒𝟕
• To calculate the MEAN add all the data points
in our sample and divide by die number of
data points (sample size).
• The MEAN can be a value that doesn’t
actually match any observation.
• The MEAN gives us useful information about
the location of our frequency distribution.
2013/05/22
2
GRAPH
0
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8 9 10
Frequency
Frequency
CALCULATE THE MEAN
Raw Data
• 𝑥 =
𝑥
𝑛
• 𝑥 is data
points
• 𝑛 is number
of
observations
Frequency
Table
• 𝑥 =
𝑥𝑓
𝑛
• 𝑥 is data
points
• 𝑛 is number
of
observations
• 𝑓 is the
frequency
Frequency
Table (Intervals)
• 𝑥 =
𝑥𝑓
𝑛
• 𝑥 is midpoints
for intervals
• 𝑛 is number
of
observations
• 𝑓 is the
frequency
CALCULATE THE MEAN - FREQUENCY TABLE:
NUBEROFFRAUDULENT CHEQUESPERWEEK
Distinct Values TallyMarks Frequency
1 / 1
2 0
3 //// 5
4 //// // 7
5 //// 4
6 //// 4
7 // 2
8 /// 3
9 // 2
10 // 2
Truck Data: weights (in tonnes) of 20 fully
loaded trucks
Truck
1
2 3 4 5 6 7 8 9 10
Weight
4.54
3.81 4.29 5.16 2.51 4.63 4.75 3.98 5.04 2.80
Truck
11
12 13 14 15 16 17 18 19 20
Weight
2.52
5.88 2.95 3.59 3.87 4.17 3.30 5.48 4.26 3.53
CALCULATE THE MEAN - GROUPED
FREQUENCY TABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
Class Intervals Frequency Midpoint
𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 𝟐. 𝟓 + 𝟑. 𝟎 ÷ 𝟐 = 2.75
𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 3.25
𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 3.75
𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 3 4.25
𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 4.75
𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 5.25
𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 5.75
MODE
•The mode is the interval with the
HIGHEST FREQUENCY.
•There can be two or more modes in a set
of data – then the mode would not be a
good measure of central tendency.
•MULTI-MODAL data consist of more than
one mode.
•UNI-MODAL data consist of only one
mode.
2013/05/22
3
GRAPH: The MODE = 4
0
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8 9 10
Frequency
Frequency
Call Centre Data: waiting times (in seconds)
for 35 randomly selected customers
C1 2 3 4 5 6 7 8 9 10 11 12
75 37 13 90 45 23 104 135 30 73 34 12
C13 14 15 16 17 18 19 20 21 22 23 24
38 40 22 47 26 57 65 33 9 85 87 16
C25 26 27 28 29 30 31 32 33 34 35
102 115 68 29 142 5 15 10 25 41 49
FREQUENCY TABLE: The MODAL CLASS is the
interval 𝟐𝟓 < 𝒙 ≤ 𝟓𝟎
Class Intervals TallyMarks Frequency
0 ≤ 𝑥 ≤ 25 //// //// 10
25 < 𝑥 ≤ 50 //// //// / 11
50 < 𝑥 ≤ 75 //// / 6
75 < 𝑥 ≤ 100 /// 3
100 < 𝑥 ≤ 125 /// 3
125 < 𝑥 ≤ 150 // 2
HISTOGRAM: MODAL CLASS (𝟐𝟓 < 𝒙 ≤ 𝟓𝟎]
0
2
4
6
8
10
12
Intervals
[0;25]
(25;50]
(50;75]
(75;100]
(100;125]
(125;150]
THE MEDIAN – RAW DATA:
Numberoffraudulentchequesreceived atabankeach weekfor30weeks
Week
1
2 3 4 5 6 7 8 9 10
5 3 8 3 3 1 10 4 6 8
Week
11
12 13 14 15 16 17 18 19 20
3 5 4 7 6 6 9 3 4 5
Week
21
22 23 24 25 26 27 28 29 30
7 9 4 5 8 6 4 4 10 4
MEDIAN
• Median = 5
• Put all observations in order from smallest to
largest, then the middle observation is the
MEDIAN.
1, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5,
5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10
2013/05/22
4
DON’T FALL INTO THE COMMON TRAP
• The median is NOT the middle of the range of
observations, for example
1, 1, 1, 1, 1, 3, 9
The median is 1 (the middle observation).
The middle of the range (9 – 1) is 5! Big
difference!
MEDIAN
Odd Number of
Observations,
for example 7
Median Position
𝒏+𝟏
𝟐
Even Number of
Observations,
for example30
Median Position
half-way between
𝒏
𝟐
𝒂𝒏𝒅 (
𝒏
𝟐
+ 𝟏)
FINDTHE MEDIAN -FREQUENCYTABLE:
NUBER OF FRAUDULENT CHEQUES PERWEEK
Distinct Values Frequency Cumulative
Frequency
1 1 1
2 0 1
3 5 6
4 7 13
5 4 17
6 4 21
7 2 23
8 3 26
9 2 28
10 2 30
FIND THE MEDIAN - GROUPED FREQUENCY
TABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
ClassIntervals Frequency Midpoint
𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 𝟐. 𝟓 + 𝟑. 𝟎 ÷ 𝟐 = 2.75
𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 3.25
𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 3.75
𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 3 4.25
𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 4.75
𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 5.25
𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 5.75
FIND THE MEDIAN FROM A GROUPED
FREQUENCY TABLE
•Median (middle observation)?
•Find the class interval in which that
observation lies.
?
CALCULATIONS
Raw Data
Mean
Mode
Median
Frequency Table
(Ungrouped
Data)
Mean
Mode
Median
Frequency Table
(Grouped Data)
Mean
Mode
Median
2013/05/22
5
HOW TO CHOOSE THE BEST MEASURE OF
LOCATION?
• When choosing the best measure of location, we
need to look as the SHAPE of the distribution.
• For nearly symmetric data, the mean is the best
choice.
• For very skewed (asymmetric) data, the mode or
median is better.
• The mean moves further along the tail than the
median, it is more sensitive to the values far from
the centre.
SYMMETRIC histogram:
Mean = Median = Mode
A POSITIVELY SKEWED (skewed to the right)
histogram has a longer tail on the right side:
Mode < Median < Mean
A NEGATIVELY SKEWED (skewed to the left)
histogram has a longer tail on the left side:
Mean < Median < Mode
PROBLEM
•We can find two very different data sets (one
distribution very spread out and another very
concentrated) with measures of central
tendency EQUAL.
•To find a true idea of our sample, we have to
MEASURE THE SPREAD OF A DISTRIBUTION,
called the spread dispersion.
MEASURESOF SPREAD(DISPERSION)
Interquartile Range
Variance
Standard Deviation
2013/05/22
6
MEASURINGSPREAD
•Think of a distribution in terms of
percentages, a horizontal axis equally divided
into 100 percentiles.
•The 10th percentile marks the point below
which 10% of the observations fall, and
above which 90% of observations fall.
•The 50th percentile, below which 50% of the
observations lie, is the median.
WORKINGWITH A PERCENTILE
• 𝑝% of the observationfall belowthe 𝑝 𝑡ℎ percentile.
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 =
𝒑
𝟏𝟎𝟎
𝒏 + 𝟏
• Workingwith the example on fraudulentcheques:
1, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6,
7, 7, 8, 8, 8, 9, 9, 10, 10
𝑷 𝟓𝟎 =
𝟓𝟎
𝟏𝟎𝟎
𝟑𝟎 + 𝟏 = 𝟏𝟓. 𝟓
• 15.5 tells us where to find our 50th percentile.
• 15 tells us which observation to go to, and 0.5 tells us how far to
move along the space between that observation and the next
highest one.
FORMULA
• 𝑷 𝟓𝟎 = 𝒙 𝟏𝟓 + 𝟎. 𝟓 𝒙 𝟏𝟔 − 𝒙 𝟏𝟓
𝑷 𝒑 = 𝒙 𝒌 + 𝒅 𝒙 𝒌+𝟏 − 𝒙 𝒌
• 𝑃 means percentile
• 𝑝 tell us which percentile
• 𝑘 the whole number calculated from the
position
• 𝑑 the decimal fraction calculated from the
position
WORKINGWITH PERCENTILESFROMUNGROUPEDFREQUENCYDATA:
NUBEROFFRAUDULENT CHEQUESPERWEEK
Distinct Values Frequency Cumulative Frequency
1 1 1
2 0 0 + 1 = 1
3 5 1 + 5 = 6
4 7 6 + 7 = 13
5 4 13 + 4 = 17
6 4 17 + 4 = 21
7 2 21 + 2 = 23
8 3 23 + 3 = 26
9 2 26 + 2 = 28
10 2 28 + 2 = 30
WORKING WITH PERCENTILES (AND
MEDIAN) FROM GROUPED DATA
• To identify the class interval 𝑳 < 𝒙 ≤ 𝑼 containing the
𝑝 𝑡ℎ percentile:
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 =
𝒑
𝟏𝟎𝟎
𝒏 + 𝟏
• The decimal fraction for grouped data is:
𝒅 =
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏−𝑺𝒖𝒎 𝒐𝒇 𝒄𝒍𝒂𝒔𝒔 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒊𝒆𝒔 𝒕𝒐 𝑳
𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒐𝒇 𝒄𝒍𝒂𝒔𝒔 𝑳 < 𝒙 ≤ 𝑼
• Calculate the 𝑝 𝑡ℎ percentile:
𝑷 𝒑 ≈ 𝑳 + 𝒅 𝑼 − 𝑳
FIND THE MEDIAN - GROUPED FREQUENCY
TABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
Class Intervals Frequency CumulativeFrequency
𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 4
𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 5
𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 10
𝟒. 𝟎 < 𝐱 ≤ 𝟒. 𝟓 3 13
𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 16
𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 19
𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 20
2013/05/22
7
FIND THEMEDIAN-GROUPEDFREQUENCYTABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
• To identify the class interval 𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 containing
the 50 𝑡ℎ percentile:
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 =
𝟓𝟎
𝟏𝟎𝟎
𝟐𝟎 + 𝟏 = 𝟏𝟎. 𝟓
• The decimal fraction for grouped data is:
𝒅 =
𝟏𝟎.𝟓 − 𝟏𝟎
𝟑
=
𝟏
𝟔
• Calculate the 𝑝 𝑡ℎ percentile:
𝑷 𝟓𝟎 ≈ 𝟒. 𝟎 + 𝒅 𝟒. 𝟓 − 𝟒. 𝟎 = 𝟒. 𝟎𝟖𝟑𝟑𝟑
MEASURINGSPREAD
• If we measure the DIFFERENCE in value between
one percentile and another, this would give us an
idea of how widely our data is spread out.
• INTERQUARTILE RANGE (IQR) = 75th – 25th Percentiles
• The bigger the IQR, the more spread out the data.
• The 75th percentile ≥ 25th percentile, therefor the
IQR ≥ 0 .
• We tend to use the MEDIAN (as measure of
central tendency) together with the IQR.
FIND THE IQR - GROUPED FREQUENCY
TABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
ClassIntervals Frequency CumulativeFrequency
𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 4
𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 5
𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 10
𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 3 13
𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 16
𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 19
𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 20
FIND THEMEDIAN-GROUPEDFREQUENCYTABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
• To identify the class interval 𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 containing
the 75 𝑡ℎ percentile:
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 =
𝟕𝟓
𝟏𝟎𝟎
𝟐𝟎 + 𝟏 = 𝟏𝟓. 𝟕𝟓
• The decimal fraction for grouped data is:
𝒅 =
𝟏𝟓. 𝟕𝟓 − 𝟏𝟑
𝟑
= 𝟎. 𝟗𝟏𝟕
• Calculate the 𝑝 𝑡ℎ percentile:
𝑷 𝟕𝟓 ≈ 𝟒. 𝟓 + 𝒅 𝟓. 𝟎 − 𝟒. 𝟓 = 𝟒. 𝟗𝟓𝟖
FIND THEMEDIAN-GROUPEDFREQUENCYTABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
• To identify the class interval 𝟑. 𝟓 < 𝒙 ≤ 𝟒.0 containing
the 25 𝑡ℎ percentile:
𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 =
𝟐𝟓
𝟏𝟎𝟎
𝟐𝟎 + 𝟏 = 𝟓. 𝟐𝟓
• The decimal fraction for grouped data is:
𝒅 =
𝟓. 𝟐𝟓 − 𝟓
𝟓
= 𝟎. 𝟎𝟓
• Calculate the 𝑝 𝑡ℎ percentile:
𝑷 𝟐𝟓 ≈ 𝟑. 𝟓 + 𝒅 𝟒. 𝟎 − 𝟑. 𝟓 = 𝟑. 𝟓𝟐𝟓
• IQR = 4.958 – 3.525 = 1.433
MEASURINGSPREAD
• When we use the MEAN as our measure of central
tendency, we usually choose A MEASURE OF HOW FAR
THE DATA IS SPREAD OUT AROUND THE MEAN.
• Two measures of spread that are based on the mean are
the VARIANCE and the STANDARD DEVIATION.
• An advantage of standard deviation is that it is measured
in the same units as the original observations.
• The variance and standard deviation are closely related.
• The variance (𝒔 𝟐 or 𝝈 𝟐) is the square of the standard
deviation (𝒔 or 𝝈).
2013/05/22
8
VARIANCE& STANDARD DEVIATION
• Variance is the rough average of all the squared
distances from the mean:
𝒔 𝟐 =
𝒙 − 𝒙 𝟐
𝒏 − 𝟏
Or
𝒔 𝟐 =
𝟏
𝒏 − 𝟏
𝒙 𝟐 −
𝒙 𝟐
𝒏
• Variance is always a positive number.
Number of fraudulent cheques received at a
bank each week for 30 weeks
Week
1
2 3 4 5 6 7 8 9 10
5 3 8 3 3 1 10 4 6 8
Week
11
12 13 14 15 16 17 18 19 20
3 5 4 7 6 6 9 3 4 5
Week
21
22 23 24 25 26 27 28 29 30
7 9 4 5 8 6 4 4 10 4
VARIANCE &STANDARD DEVIATIONFROMRAWDATA 𝒙 = 𝟓. 𝟒𝟕
Distinct
Values
𝒙 − 𝒙 𝒙 − 𝒙 𝟐 Frequencies
𝒇 𝒙 − 𝒙 𝟐
1 1 − 5.47 = −4.47 −4.47 2
= 19.9809 𝟏𝟗. 𝟗𝟖𝟎𝟗
2 −3.47 12.0409 𝟎
3 −2.47 6.1009 𝟑𝟎. 𝟓𝟎𝟒𝟓
4 −1.47 2.1609 𝟏𝟓. 𝟏𝟐𝟔𝟑
5 0.47 0.2209 𝟎. 𝟖𝟖𝟑𝟔
6 0.53 0.2809 𝟏. 𝟏𝟐𝟑𝟔
7 1.53 2.3409 𝟒. 𝟔𝟖𝟏𝟖
8 2.53 6.4009 𝟏𝟗. 𝟐𝟎𝟐𝟕
9 3.53 12.4609 𝟐𝟒. 𝟗𝟐𝟏𝟖
10 4.53 20.5209 𝟒𝟏. 𝟎𝟒𝟏𝟖
(𝒙 − 𝒙 ) = 0 𝒙 − 𝒙 𝟐 = 82.509
𝟏𝟓𝟕. 𝟒𝟔𝟕
CALCULATE THE VARIANCE &STANDARD DEVIATION -
FREQUENCY TABLE:
NUBEROFFRAUDULENT CHEQUESPERWEEK
Distinct Values Frequency Squared Observation
1 1 1
2 0 4
3 5 9
4 7 16
5 4 25
6 4 36
7 2 49
8 3 64
9 2 81
10 2 100
VARIANCE & STANDARD DEVIATION FROM
UNGROUPED FREQUENCY DATA
𝒔 𝟐
=
𝟏
𝒏 − 𝟏
𝒇𝒙 𝟐
−
𝒇𝒙 𝟐
𝒏
• Variance:
𝒔 𝟐
=
𝟏
𝟑𝟎 − 𝟏
𝟏𝟎𝟓𝟒 −
𝟏𝟔𝟒 𝟐
𝟑𝟎
= 𝟓. 𝟒𝟐𝟗𝟗
• Standard deviation: 𝑠 = 𝜎 = 5.4299 = 𝟐. 𝟑𝟑
FIND THE VARIANCE & STANDARD
DEVIATION - GROUPED FREQUENCY TABLE:
TruckData: weights(intonnes)of20fullyloadedtrucks
Class Intervals Frequency Midpoint Squared Midpoint
𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 2.75 7.5625
𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 3.25 10.5625
𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 3.75 14.0625
𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 3 4.25 18.0625
𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 4.75 22.5625
𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 5.25 27.5625
𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 5.75 33.0625
2013/05/22
9
VARIANCE & STANDARD DEVIATION FROM
GROUPED DATA
𝒔 𝟐
=
𝟏
𝒏 − 𝟏
𝒇𝒙 𝟐
−
𝒇𝒙 𝟐
𝒏
• Variance:
𝒔 𝟐
=
𝟏
𝟐𝟎 − 𝟏
𝟑𝟒𝟖. 𝟕𝟓 −
𝟖𝟏. 𝟓 𝟐
𝟐𝟎
= 𝟎. 𝟖𝟕𝟓𝟕
• Standard deviation: 𝑠 = 𝜎 = 0.8757 = 𝟎. 𝟗𝟒
CALCULATIONS
Raw Data
IQR
Variance &
Standard
Deviation
Frequency Table
(Ungrouped
Data)
IQR
Variance &
Standard
Deviation
Frequency Table
(Grouped Data)
IQR
Variance &
Standard
Deviation
BOX - AND - WISKER DIAGRAM
(5 POINT SUMMARY)
Minimum
Value
𝑸 𝟏 = 𝑷 𝟐𝟓
Median𝑸 𝟑 = 𝑷 𝟕𝟓
Maximum
Value
EXAMPLE
Consider the following set of 23 scores:
0 3 4 8 9 12 14 15 16 16 16 18
19 21 22 25 32 34 39 43 54 67 77
1. Find the 5 point summary
2. Draw a box – and – wisher plot to
illustrate the values
5 - POINT SUMMARY
0 3 4 8 9 12 14 15 16 16 16 18
19 21 22 25 32 34 39 43 54 67 77
HOMEWORK
•Example X-Kit textbook page 218 – 223.
•“Practise for your exams” page 224
number 1 & 2.
•Par B.2 (page B5) all odd number
questions.

Mais conteúdo relacionado

Mais procurados

Measures of Position - Elementary Statistics
Measures of Position - Elementary StatisticsMeasures of Position - Elementary Statistics
Measures of Position - Elementary StatisticsFlipped Channel
 
Measures of-central-tendency-dispersion
Measures of-central-tendency-dispersionMeasures of-central-tendency-dispersion
Measures of-central-tendency-dispersionSanoj Fernando
 
Unit iii measures of dispersion (2)
Unit iii  measures of dispersion (2)Unit iii  measures of dispersion (2)
Unit iii measures of dispersion (2)Sanoj Fernando
 
2 6 measure of position
2 6 measure of position2 6 measure of position
2 6 measure of positionKen Kretsch
 
introduction to biostat, standard deviation and variance
introduction to biostat, standard deviation and varianceintroduction to biostat, standard deviation and variance
introduction to biostat, standard deviation and varianceamol askar
 
Normal Curve and Standard Scores
Normal Curve and Standard ScoresNormal Curve and Standard Scores
Normal Curve and Standard ScoresJenewel Azuelo
 
M.Ed Tcs 2 seminar ppt npc to submit
M.Ed Tcs 2 seminar ppt npc   to submitM.Ed Tcs 2 seminar ppt npc   to submit
M.Ed Tcs 2 seminar ppt npc to submitBINCYKMATHEW
 
Lesson03_new
Lesson03_newLesson03_new
Lesson03_newshengvn
 
Normal distribution
Normal distributionNormal distribution
Normal distributionGlobal Polis
 
Normal distribution curve
Normal distribution curveNormal distribution curve
Normal distribution curveFahadi302
 
3.1-3.2 Measures of Central Tendency
3.1-3.2 Measures of Central Tendency3.1-3.2 Measures of Central Tendency
3.1-3.2 Measures of Central Tendencymlong24
 
3.4 Measures of Position
3.4 Measures of Position3.4 Measures of Position
3.4 Measures of Positionmlong24
 
Lesson 7 measures of dispersion part 2
Lesson 7 measures of dispersion part 2Lesson 7 measures of dispersion part 2
Lesson 7 measures of dispersion part 2nurun2010
 

Mais procurados (20)

Measures of Position - Elementary Statistics
Measures of Position - Elementary StatisticsMeasures of Position - Elementary Statistics
Measures of Position - Elementary Statistics
 
Dispersion
DispersionDispersion
Dispersion
 
Measures of-central-tendency-dispersion
Measures of-central-tendency-dispersionMeasures of-central-tendency-dispersion
Measures of-central-tendency-dispersion
 
Unit iii measures of dispersion (2)
Unit iii  measures of dispersion (2)Unit iii  measures of dispersion (2)
Unit iii measures of dispersion (2)
 
2 6 measure of position
2 6 measure of position2 6 measure of position
2 6 measure of position
 
introduction to biostat, standard deviation and variance
introduction to biostat, standard deviation and varianceintroduction to biostat, standard deviation and variance
introduction to biostat, standard deviation and variance
 
Descriptive statistics -review(2)
Descriptive statistics -review(2)Descriptive statistics -review(2)
Descriptive statistics -review(2)
 
Normal Curve and Standard Scores
Normal Curve and Standard ScoresNormal Curve and Standard Scores
Normal Curve and Standard Scores
 
Descriptive Statistics
Descriptive StatisticsDescriptive Statistics
Descriptive Statistics
 
M.Ed Tcs 2 seminar ppt npc to submit
M.Ed Tcs 2 seminar ppt npc   to submitM.Ed Tcs 2 seminar ppt npc   to submit
M.Ed Tcs 2 seminar ppt npc to submit
 
Lesson03_new
Lesson03_newLesson03_new
Lesson03_new
 
Normal distribution
Normal distributionNormal distribution
Normal distribution
 
Measures of dispersion discuss 2.2
Measures of dispersion discuss 2.2Measures of dispersion discuss 2.2
Measures of dispersion discuss 2.2
 
Chapter03
Chapter03Chapter03
Chapter03
 
Normal distribution curve
Normal distribution curveNormal distribution curve
Normal distribution curve
 
3.1-3.2 Measures of Central Tendency
3.1-3.2 Measures of Central Tendency3.1-3.2 Measures of Central Tendency
3.1-3.2 Measures of Central Tendency
 
3.4 Measures of Position
3.4 Measures of Position3.4 Measures of Position
3.4 Measures of Position
 
Summarizing data
Summarizing dataSummarizing data
Summarizing data
 
Measures of central tendency
Measures of central tendencyMeasures of central tendency
Measures of central tendency
 
Lesson 7 measures of dispersion part 2
Lesson 7 measures of dispersion part 2Lesson 7 measures of dispersion part 2
Lesson 7 measures of dispersion part 2
 

Destaque

Introduce Node.js Taiwan community
Introduce Node.js Taiwan communityIntroduce Node.js Taiwan community
Introduce Node.js Taiwan communityCaesar Chi
 
Resintencia, mediciones y codigo de colores
Resintencia, mediciones y codigo de coloresResintencia, mediciones y codigo de colores
Resintencia, mediciones y codigo de coloresJairQQ
 
Presentation1
Presentation1Presentation1
Presentation1Lu Jiaqi
 
三個鞠躬1030502cms
三個鞠躬1030502cms三個鞠躬1030502cms
三個鞠躬1030502cmsPhil Wen
 
La organizacion administrativa del estado mexicano
La organizacion administrativa del estado mexicanoLa organizacion administrativa del estado mexicano
La organizacion administrativa del estado mexicanoDaniel Garcia
 
AnaClaudiaAlmeidaTaveira
AnaClaudiaAlmeidaTaveiraAnaClaudiaAlmeidaTaveira
AnaClaudiaAlmeidaTaveiraAna Taveira
 
London Dine & Wine- A Bloomberg Brief Special Supplement
London Dine & Wine- A Bloomberg Brief Special Supplement London Dine & Wine- A Bloomberg Brief Special Supplement
London Dine & Wine- A Bloomberg Brief Special Supplement Bloomberg Briefs
 
CPD Newsletter, January-March 2016
CPD Newsletter, January-March 2016CPD Newsletter, January-March 2016
CPD Newsletter, January-March 2016Sazzad Mahmud Shuvo
 
The 9 Circles of Employee Engagement Hell
The 9 Circles of Employee Engagement Hell The 9 Circles of Employee Engagement Hell
The 9 Circles of Employee Engagement Hell Globoforce
 
Roland Xp-10 service manual keyboard
Roland Xp-10 service manual keyboardRoland Xp-10 service manual keyboard
Roland Xp-10 service manual keyboardQuiller123
 
Pinstripe Presents: Sharing Your Talent Mindset
Pinstripe Presents: Sharing Your Talent MindsetPinstripe Presents: Sharing Your Talent Mindset
Pinstripe Presents: Sharing Your Talent MindsetCielo
 

Destaque (18)

Introduce Node.js Taiwan community
Introduce Node.js Taiwan communityIntroduce Node.js Taiwan community
Introduce Node.js Taiwan community
 
Resintencia, mediciones y codigo de colores
Resintencia, mediciones y codigo de coloresResintencia, mediciones y codigo de colores
Resintencia, mediciones y codigo de colores
 
Presentation1
Presentation1Presentation1
Presentation1
 
MÍDIA KIT - CHARLES ARAUJO
MÍDIA KIT - CHARLES ARAUJOMÍDIA KIT - CHARLES ARAUJO
MÍDIA KIT - CHARLES ARAUJO
 
三個鞠躬1030502cms
三個鞠躬1030502cms三個鞠躬1030502cms
三個鞠躬1030502cms
 
Shockley ppt ch12
Shockley ppt ch12Shockley ppt ch12
Shockley ppt ch12
 
La organizacion administrativa del estado mexicano
La organizacion administrativa del estado mexicanoLa organizacion administrativa del estado mexicano
La organizacion administrativa del estado mexicano
 
AnaClaudiaAlmeidaTaveira
AnaClaudiaAlmeidaTaveiraAnaClaudiaAlmeidaTaveira
AnaClaudiaAlmeidaTaveira
 
London Dine & Wine- A Bloomberg Brief Special Supplement
London Dine & Wine- A Bloomberg Brief Special Supplement London Dine & Wine- A Bloomberg Brief Special Supplement
London Dine & Wine- A Bloomberg Brief Special Supplement
 
CPD Newsletter, January-March 2016
CPD Newsletter, January-March 2016CPD Newsletter, January-March 2016
CPD Newsletter, January-March 2016
 
¿Que sabe Ud. de nutrición?
¿Que sabe Ud. de nutrición?¿Que sabe Ud. de nutrición?
¿Que sabe Ud. de nutrición?
 
The 9 Circles of Employee Engagement Hell
The 9 Circles of Employee Engagement Hell The 9 Circles of Employee Engagement Hell
The 9 Circles of Employee Engagement Hell
 
The Role of Outreach?
The Role of Outreach?The Role of Outreach?
The Role of Outreach?
 
What is usability
What is usabilityWhat is usability
What is usability
 
Roland Xp-10 service manual keyboard
Roland Xp-10 service manual keyboardRoland Xp-10 service manual keyboard
Roland Xp-10 service manual keyboard
 
Ethics Commission Training version 4
Ethics Commission Training version 4Ethics Commission Training version 4
Ethics Commission Training version 4
 
Pinstripe Presents: Sharing Your Talent Mindset
Pinstripe Presents: Sharing Your Talent MindsetPinstripe Presents: Sharing Your Talent Mindset
Pinstripe Presents: Sharing Your Talent Mindset
 
Arpwatch
ArpwatchArpwatch
Arpwatch
 

Semelhante a Statistics

maft0a2_Statistics_lecture2_2021.pptx
maft0a2_Statistics_lecture2_2021.pptxmaft0a2_Statistics_lecture2_2021.pptx
maft0a2_Statistics_lecture2_2021.pptxTshegofatso Mphake
 
2.-Measures-of-central-tendency.pdf assessment in learning 2
2.-Measures-of-central-tendency.pdf assessment in learning 22.-Measures-of-central-tendency.pdf assessment in learning 2
2.-Measures-of-central-tendency.pdf assessment in learning 2aprilanngastador165
 
L1 statistics
L1 statisticsL1 statistics
L1 statisticsdapdai
 
Chapter 3 Summary Measures_Complete-1.pdf
Chapter 3 Summary Measures_Complete-1.pdfChapter 3 Summary Measures_Complete-1.pdf
Chapter 3 Summary Measures_Complete-1.pdfCalhyneJose
 
Business Statistics Chapter 2
Business Statistics Chapter 2Business Statistics Chapter 2
Business Statistics Chapter 2Lux PP
 
Estimating Tail Parameters
Estimating Tail ParametersEstimating Tail Parameters
Estimating Tail ParametersAlejandro Ortega
 
Biostatistics Measures of central tendency
Biostatistics Measures of central tendency Biostatistics Measures of central tendency
Biostatistics Measures of central tendency HARINATHA REDDY ASWARTHA
 
measure of dispersion
measure of dispersion measure of dispersion
measure of dispersion som allul
 
Measure of Variability Report.pptx
Measure of Variability Report.pptxMeasure of Variability Report.pptx
Measure of Variability Report.pptxCalvinAdorDionisio
 
Missing Parts I don’t think you understood the assignment.docx
Missing Parts I don’t think you understood the assignment.docxMissing Parts I don’t think you understood the assignment.docx
Missing Parts I don’t think you understood the assignment.docxannandleola
 
Lect 3 background mathematics for Data Mining
Lect 3 background mathematics for Data MiningLect 3 background mathematics for Data Mining
Lect 3 background mathematics for Data Mininghktripathy
 
Lect 3 background mathematics
Lect 3 background mathematicsLect 3 background mathematics
Lect 3 background mathematicshktripathy
 
analytical representation of data
 analytical representation of data analytical representation of data
analytical representation of dataUnsa Shakir
 
3A. MEASURES OF CENTRAL TENDENCY UNGROUP AND GROUP DATA.pptx
3A. MEASURES OF CENTRAL TENDENCY UNGROUP AND GROUP DATA.pptx3A. MEASURES OF CENTRAL TENDENCY UNGROUP AND GROUP DATA.pptx
3A. MEASURES OF CENTRAL TENDENCY UNGROUP AND GROUP DATA.pptxKarenKayeJimenez2
 
3. Descriptive statistics.pdf
3. Descriptive statistics.pdf3. Descriptive statistics.pdf
3. Descriptive statistics.pdfYomifDeksisaHerpa
 
Application of Machine Learning in Agriculture
Application of Machine  Learning in AgricultureApplication of Machine  Learning in Agriculture
Application of Machine Learning in AgricultureAman Vasisht
 

Semelhante a Statistics (20)

maft0a2_Statistics_lecture2_2021.pptx
maft0a2_Statistics_lecture2_2021.pptxmaft0a2_Statistics_lecture2_2021.pptx
maft0a2_Statistics_lecture2_2021.pptx
 
2.-Measures-of-central-tendency.pdf assessment in learning 2
2.-Measures-of-central-tendency.pdf assessment in learning 22.-Measures-of-central-tendency.pdf assessment in learning 2
2.-Measures-of-central-tendency.pdf assessment in learning 2
 
L1 statistics
L1 statisticsL1 statistics
L1 statistics
 
Chapter 3 Summary Measures_Complete-1.pdf
Chapter 3 Summary Measures_Complete-1.pdfChapter 3 Summary Measures_Complete-1.pdf
Chapter 3 Summary Measures_Complete-1.pdf
 
Measures-of-Central-Tendency.ppt
Measures-of-Central-Tendency.pptMeasures-of-Central-Tendency.ppt
Measures-of-Central-Tendency.ppt
 
Business Statistics Chapter 2
Business Statistics Chapter 2Business Statistics Chapter 2
Business Statistics Chapter 2
 
DescriptiveStatistics.pdf
DescriptiveStatistics.pdfDescriptiveStatistics.pdf
DescriptiveStatistics.pdf
 
Estimating Tail Parameters
Estimating Tail ParametersEstimating Tail Parameters
Estimating Tail Parameters
 
Biostatistics Measures of central tendency
Biostatistics Measures of central tendency Biostatistics Measures of central tendency
Biostatistics Measures of central tendency
 
measure of dispersion
measure of dispersion measure of dispersion
measure of dispersion
 
Measure of Variability Report.pptx
Measure of Variability Report.pptxMeasure of Variability Report.pptx
Measure of Variability Report.pptx
 
Missing Parts I don’t think you understood the assignment.docx
Missing Parts I don’t think you understood the assignment.docxMissing Parts I don’t think you understood the assignment.docx
Missing Parts I don’t think you understood the assignment.docx
 
Lect 3 background mathematics for Data Mining
Lect 3 background mathematics for Data MiningLect 3 background mathematics for Data Mining
Lect 3 background mathematics for Data Mining
 
Lect 3 background mathematics
Lect 3 background mathematicsLect 3 background mathematics
Lect 3 background mathematics
 
Qc tools
Qc toolsQc tools
Qc tools
 
Qc tools
Qc toolsQc tools
Qc tools
 
analytical representation of data
 analytical representation of data analytical representation of data
analytical representation of data
 
3A. MEASURES OF CENTRAL TENDENCY UNGROUP AND GROUP DATA.pptx
3A. MEASURES OF CENTRAL TENDENCY UNGROUP AND GROUP DATA.pptx3A. MEASURES OF CENTRAL TENDENCY UNGROUP AND GROUP DATA.pptx
3A. MEASURES OF CENTRAL TENDENCY UNGROUP AND GROUP DATA.pptx
 
3. Descriptive statistics.pdf
3. Descriptive statistics.pdf3. Descriptive statistics.pdf
3. Descriptive statistics.pdf
 
Application of Machine Learning in Agriculture
Application of Machine  Learning in AgricultureApplication of Machine  Learning in Agriculture
Application of Machine Learning in Agriculture
 

Último

PISA-VET launch_El Iza Mohamedou_19 March 2024.pptx
PISA-VET launch_El Iza Mohamedou_19 March 2024.pptxPISA-VET launch_El Iza Mohamedou_19 March 2024.pptx
PISA-VET launch_El Iza Mohamedou_19 March 2024.pptxEduSkills OECD
 
General views of Histopathology and step
General views of Histopathology and stepGeneral views of Histopathology and step
General views of Histopathology and stepobaje godwin sunday
 
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptxClinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptxraviapr7
 
Ultra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxUltra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxDr. Asif Anas
 
Prescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxPrescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxraviapr7
 
Education and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxEducation and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxraviapr7
 
In - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptxIn - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptxAditiChauhan701637
 
How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17Celine George
 
The basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxThe basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxheathfieldcps1
 
Benefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationBenefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationMJDuyan
 
Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.raviapr7
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxiammrhaywood
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17Celine George
 
How to Use api.constrains ( ) in Odoo 17
How to Use api.constrains ( ) in Odoo 17How to Use api.constrains ( ) in Odoo 17
How to Use api.constrains ( ) in Odoo 17Celine George
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17Celine George
 
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRADUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRATanmoy Mishra
 
3.21.24 The Origins of Black Power.pptx
3.21.24  The Origins of Black Power.pptx3.21.24  The Origins of Black Power.pptx
3.21.24 The Origins of Black Power.pptxmary850239
 
Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...raviapr7
 
Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.EnglishCEIPdeSigeiro
 
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdf
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdfMaximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdf
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdfTechSoup
 

Último (20)

PISA-VET launch_El Iza Mohamedou_19 March 2024.pptx
PISA-VET launch_El Iza Mohamedou_19 March 2024.pptxPISA-VET launch_El Iza Mohamedou_19 March 2024.pptx
PISA-VET launch_El Iza Mohamedou_19 March 2024.pptx
 
General views of Histopathology and step
General views of Histopathology and stepGeneral views of Histopathology and step
General views of Histopathology and step
 
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptxClinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
 
Ultra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxUltra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptx
 
Prescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxPrescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptx
 
Education and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxEducation and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptx
 
In - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptxIn - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptx
 
How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17How to Make a Field read-only in Odoo 17
How to Make a Field read-only in Odoo 17
 
The basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxThe basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptx
 
Benefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationBenefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive Education
 
Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17
 
How to Use api.constrains ( ) in Odoo 17
How to Use api.constrains ( ) in Odoo 17How to Use api.constrains ( ) in Odoo 17
How to Use api.constrains ( ) in Odoo 17
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17
 
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRADUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
 
3.21.24 The Origins of Black Power.pptx
3.21.24  The Origins of Black Power.pptx3.21.24  The Origins of Black Power.pptx
3.21.24 The Origins of Black Power.pptx
 
Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...
 
Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.
 
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdf
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdfMaximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdf
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdf
 

Statistics

  • 1. 2013/05/22 1 STATISTICS X-Kit Textbook Chapter 9 Precalculus Textbook Appendix B: Concepts in Statistics Par B.2 CONTENT THE GOAL Look at ways of summarising a large amount of sample data in just one or two key numbers. Two important aspects of a set of data: •The LOCATION •The SPREAD MEASURES OF CENTRAL TENDENCY (LOCATION) Arithmetic Mean (Average) Mode (the highest point/frequency) Median (the middle observation) Number of fraudulent cheques received at a bank each week for 30 weeks Week 1 2 3 4 5 6 7 8 9 10 5 3 8 3 3 1 10 4 6 8 Week 11 12 13 14 15 16 17 18 19 20 3 5 4 7 6 6 9 3 4 5 Week 21 22 23 24 25 26 27 28 29 30 7 9 4 5 8 6 4 4 10 4 ARITHMETIC MEAN • 𝒙 = 𝟏𝟔𝟒 𝟑𝟎 = 𝟓. 𝟒𝟕 • To calculate the MEAN add all the data points in our sample and divide by die number of data points (sample size). • The MEAN can be a value that doesn’t actually match any observation. • The MEAN gives us useful information about the location of our frequency distribution.
  • 2. 2013/05/22 2 GRAPH 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 Frequency Frequency CALCULATE THE MEAN Raw Data • 𝑥 = 𝑥 𝑛 • 𝑥 is data points • 𝑛 is number of observations Frequency Table • 𝑥 = 𝑥𝑓 𝑛 • 𝑥 is data points • 𝑛 is number of observations • 𝑓 is the frequency Frequency Table (Intervals) • 𝑥 = 𝑥𝑓 𝑛 • 𝑥 is midpoints for intervals • 𝑛 is number of observations • 𝑓 is the frequency CALCULATE THE MEAN - FREQUENCY TABLE: NUBEROFFRAUDULENT CHEQUESPERWEEK Distinct Values TallyMarks Frequency 1 / 1 2 0 3 //// 5 4 //// // 7 5 //// 4 6 //// 4 7 // 2 8 /// 3 9 // 2 10 // 2 Truck Data: weights (in tonnes) of 20 fully loaded trucks Truck 1 2 3 4 5 6 7 8 9 10 Weight 4.54 3.81 4.29 5.16 2.51 4.63 4.75 3.98 5.04 2.80 Truck 11 12 13 14 15 16 17 18 19 20 Weight 2.52 5.88 2.95 3.59 3.87 4.17 3.30 5.48 4.26 3.53 CALCULATE THE MEAN - GROUPED FREQUENCY TABLE: TruckData: weights(intonnes)of20fullyloadedtrucks Class Intervals Frequency Midpoint 𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 𝟐. 𝟓 + 𝟑. 𝟎 ÷ 𝟐 = 2.75 𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 3.25 𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 3.75 𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 3 4.25 𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 4.75 𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 5.25 𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 5.75 MODE •The mode is the interval with the HIGHEST FREQUENCY. •There can be two or more modes in a set of data – then the mode would not be a good measure of central tendency. •MULTI-MODAL data consist of more than one mode. •UNI-MODAL data consist of only one mode.
  • 3. 2013/05/22 3 GRAPH: The MODE = 4 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 Frequency Frequency Call Centre Data: waiting times (in seconds) for 35 randomly selected customers C1 2 3 4 5 6 7 8 9 10 11 12 75 37 13 90 45 23 104 135 30 73 34 12 C13 14 15 16 17 18 19 20 21 22 23 24 38 40 22 47 26 57 65 33 9 85 87 16 C25 26 27 28 29 30 31 32 33 34 35 102 115 68 29 142 5 15 10 25 41 49 FREQUENCY TABLE: The MODAL CLASS is the interval 𝟐𝟓 < 𝒙 ≤ 𝟓𝟎 Class Intervals TallyMarks Frequency 0 ≤ 𝑥 ≤ 25 //// //// 10 25 < 𝑥 ≤ 50 //// //// / 11 50 < 𝑥 ≤ 75 //// / 6 75 < 𝑥 ≤ 100 /// 3 100 < 𝑥 ≤ 125 /// 3 125 < 𝑥 ≤ 150 // 2 HISTOGRAM: MODAL CLASS (𝟐𝟓 < 𝒙 ≤ 𝟓𝟎] 0 2 4 6 8 10 12 Intervals [0;25] (25;50] (50;75] (75;100] (100;125] (125;150] THE MEDIAN – RAW DATA: Numberoffraudulentchequesreceived atabankeach weekfor30weeks Week 1 2 3 4 5 6 7 8 9 10 5 3 8 3 3 1 10 4 6 8 Week 11 12 13 14 15 16 17 18 19 20 3 5 4 7 6 6 9 3 4 5 Week 21 22 23 24 25 26 27 28 29 30 7 9 4 5 8 6 4 4 10 4 MEDIAN • Median = 5 • Put all observations in order from smallest to largest, then the middle observation is the MEDIAN. 1, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10
  • 4. 2013/05/22 4 DON’T FALL INTO THE COMMON TRAP • The median is NOT the middle of the range of observations, for example 1, 1, 1, 1, 1, 3, 9 The median is 1 (the middle observation). The middle of the range (9 – 1) is 5! Big difference! MEDIAN Odd Number of Observations, for example 7 Median Position 𝒏+𝟏 𝟐 Even Number of Observations, for example30 Median Position half-way between 𝒏 𝟐 𝒂𝒏𝒅 ( 𝒏 𝟐 + 𝟏) FINDTHE MEDIAN -FREQUENCYTABLE: NUBER OF FRAUDULENT CHEQUES PERWEEK Distinct Values Frequency Cumulative Frequency 1 1 1 2 0 1 3 5 6 4 7 13 5 4 17 6 4 21 7 2 23 8 3 26 9 2 28 10 2 30 FIND THE MEDIAN - GROUPED FREQUENCY TABLE: TruckData: weights(intonnes)of20fullyloadedtrucks ClassIntervals Frequency Midpoint 𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 𝟐. 𝟓 + 𝟑. 𝟎 ÷ 𝟐 = 2.75 𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 3.25 𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 3.75 𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 3 4.25 𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 4.75 𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 5.25 𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 5.75 FIND THE MEDIAN FROM A GROUPED FREQUENCY TABLE •Median (middle observation)? •Find the class interval in which that observation lies. ? CALCULATIONS Raw Data Mean Mode Median Frequency Table (Ungrouped Data) Mean Mode Median Frequency Table (Grouped Data) Mean Mode Median
  • 5. 2013/05/22 5 HOW TO CHOOSE THE BEST MEASURE OF LOCATION? • When choosing the best measure of location, we need to look as the SHAPE of the distribution. • For nearly symmetric data, the mean is the best choice. • For very skewed (asymmetric) data, the mode or median is better. • The mean moves further along the tail than the median, it is more sensitive to the values far from the centre. SYMMETRIC histogram: Mean = Median = Mode A POSITIVELY SKEWED (skewed to the right) histogram has a longer tail on the right side: Mode < Median < Mean A NEGATIVELY SKEWED (skewed to the left) histogram has a longer tail on the left side: Mean < Median < Mode PROBLEM •We can find two very different data sets (one distribution very spread out and another very concentrated) with measures of central tendency EQUAL. •To find a true idea of our sample, we have to MEASURE THE SPREAD OF A DISTRIBUTION, called the spread dispersion. MEASURESOF SPREAD(DISPERSION) Interquartile Range Variance Standard Deviation
  • 6. 2013/05/22 6 MEASURINGSPREAD •Think of a distribution in terms of percentages, a horizontal axis equally divided into 100 percentiles. •The 10th percentile marks the point below which 10% of the observations fall, and above which 90% of observations fall. •The 50th percentile, below which 50% of the observations lie, is the median. WORKINGWITH A PERCENTILE • 𝑝% of the observationfall belowthe 𝑝 𝑡ℎ percentile. 𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 = 𝒑 𝟏𝟎𝟎 𝒏 + 𝟏 • Workingwith the example on fraudulentcheques: 1, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10 𝑷 𝟓𝟎 = 𝟓𝟎 𝟏𝟎𝟎 𝟑𝟎 + 𝟏 = 𝟏𝟓. 𝟓 • 15.5 tells us where to find our 50th percentile. • 15 tells us which observation to go to, and 0.5 tells us how far to move along the space between that observation and the next highest one. FORMULA • 𝑷 𝟓𝟎 = 𝒙 𝟏𝟓 + 𝟎. 𝟓 𝒙 𝟏𝟔 − 𝒙 𝟏𝟓 𝑷 𝒑 = 𝒙 𝒌 + 𝒅 𝒙 𝒌+𝟏 − 𝒙 𝒌 • 𝑃 means percentile • 𝑝 tell us which percentile • 𝑘 the whole number calculated from the position • 𝑑 the decimal fraction calculated from the position WORKINGWITH PERCENTILESFROMUNGROUPEDFREQUENCYDATA: NUBEROFFRAUDULENT CHEQUESPERWEEK Distinct Values Frequency Cumulative Frequency 1 1 1 2 0 0 + 1 = 1 3 5 1 + 5 = 6 4 7 6 + 7 = 13 5 4 13 + 4 = 17 6 4 17 + 4 = 21 7 2 21 + 2 = 23 8 3 23 + 3 = 26 9 2 26 + 2 = 28 10 2 28 + 2 = 30 WORKING WITH PERCENTILES (AND MEDIAN) FROM GROUPED DATA • To identify the class interval 𝑳 < 𝒙 ≤ 𝑼 containing the 𝑝 𝑡ℎ percentile: 𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 = 𝒑 𝟏𝟎𝟎 𝒏 + 𝟏 • The decimal fraction for grouped data is: 𝒅 = 𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏−𝑺𝒖𝒎 𝒐𝒇 𝒄𝒍𝒂𝒔𝒔 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒊𝒆𝒔 𝒕𝒐 𝑳 𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒐𝒇 𝒄𝒍𝒂𝒔𝒔 𝑳 < 𝒙 ≤ 𝑼 • Calculate the 𝑝 𝑡ℎ percentile: 𝑷 𝒑 ≈ 𝑳 + 𝒅 𝑼 − 𝑳 FIND THE MEDIAN - GROUPED FREQUENCY TABLE: TruckData: weights(intonnes)of20fullyloadedtrucks Class Intervals Frequency CumulativeFrequency 𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 4 𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 5 𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 10 𝟒. 𝟎 < 𝐱 ≤ 𝟒. 𝟓 3 13 𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 16 𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 19 𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 20
  • 7. 2013/05/22 7 FIND THEMEDIAN-GROUPEDFREQUENCYTABLE: TruckData: weights(intonnes)of20fullyloadedtrucks • To identify the class interval 𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 containing the 50 𝑡ℎ percentile: 𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 = 𝟓𝟎 𝟏𝟎𝟎 𝟐𝟎 + 𝟏 = 𝟏𝟎. 𝟓 • The decimal fraction for grouped data is: 𝒅 = 𝟏𝟎.𝟓 − 𝟏𝟎 𝟑 = 𝟏 𝟔 • Calculate the 𝑝 𝑡ℎ percentile: 𝑷 𝟓𝟎 ≈ 𝟒. 𝟎 + 𝒅 𝟒. 𝟓 − 𝟒. 𝟎 = 𝟒. 𝟎𝟖𝟑𝟑𝟑 MEASURINGSPREAD • If we measure the DIFFERENCE in value between one percentile and another, this would give us an idea of how widely our data is spread out. • INTERQUARTILE RANGE (IQR) = 75th – 25th Percentiles • The bigger the IQR, the more spread out the data. • The 75th percentile ≥ 25th percentile, therefor the IQR ≥ 0 . • We tend to use the MEDIAN (as measure of central tendency) together with the IQR. FIND THE IQR - GROUPED FREQUENCY TABLE: TruckData: weights(intonnes)of20fullyloadedtrucks ClassIntervals Frequency CumulativeFrequency 𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 4 𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 5 𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 10 𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 3 13 𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 16 𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 19 𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 20 FIND THEMEDIAN-GROUPEDFREQUENCYTABLE: TruckData: weights(intonnes)of20fullyloadedtrucks • To identify the class interval 𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 containing the 75 𝑡ℎ percentile: 𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 = 𝟕𝟓 𝟏𝟎𝟎 𝟐𝟎 + 𝟏 = 𝟏𝟓. 𝟕𝟓 • The decimal fraction for grouped data is: 𝒅 = 𝟏𝟓. 𝟕𝟓 − 𝟏𝟑 𝟑 = 𝟎. 𝟗𝟏𝟕 • Calculate the 𝑝 𝑡ℎ percentile: 𝑷 𝟕𝟓 ≈ 𝟒. 𝟓 + 𝒅 𝟓. 𝟎 − 𝟒. 𝟓 = 𝟒. 𝟗𝟓𝟖 FIND THEMEDIAN-GROUPEDFREQUENCYTABLE: TruckData: weights(intonnes)of20fullyloadedtrucks • To identify the class interval 𝟑. 𝟓 < 𝒙 ≤ 𝟒.0 containing the 25 𝑡ℎ percentile: 𝑷𝒐𝒔𝒊𝒕𝒊𝒐𝒏 = 𝟐𝟓 𝟏𝟎𝟎 𝟐𝟎 + 𝟏 = 𝟓. 𝟐𝟓 • The decimal fraction for grouped data is: 𝒅 = 𝟓. 𝟐𝟓 − 𝟓 𝟓 = 𝟎. 𝟎𝟓 • Calculate the 𝑝 𝑡ℎ percentile: 𝑷 𝟐𝟓 ≈ 𝟑. 𝟓 + 𝒅 𝟒. 𝟎 − 𝟑. 𝟓 = 𝟑. 𝟓𝟐𝟓 • IQR = 4.958 – 3.525 = 1.433 MEASURINGSPREAD • When we use the MEAN as our measure of central tendency, we usually choose A MEASURE OF HOW FAR THE DATA IS SPREAD OUT AROUND THE MEAN. • Two measures of spread that are based on the mean are the VARIANCE and the STANDARD DEVIATION. • An advantage of standard deviation is that it is measured in the same units as the original observations. • The variance and standard deviation are closely related. • The variance (𝒔 𝟐 or 𝝈 𝟐) is the square of the standard deviation (𝒔 or 𝝈).
  • 8. 2013/05/22 8 VARIANCE& STANDARD DEVIATION • Variance is the rough average of all the squared distances from the mean: 𝒔 𝟐 = 𝒙 − 𝒙 𝟐 𝒏 − 𝟏 Or 𝒔 𝟐 = 𝟏 𝒏 − 𝟏 𝒙 𝟐 − 𝒙 𝟐 𝒏 • Variance is always a positive number. Number of fraudulent cheques received at a bank each week for 30 weeks Week 1 2 3 4 5 6 7 8 9 10 5 3 8 3 3 1 10 4 6 8 Week 11 12 13 14 15 16 17 18 19 20 3 5 4 7 6 6 9 3 4 5 Week 21 22 23 24 25 26 27 28 29 30 7 9 4 5 8 6 4 4 10 4 VARIANCE &STANDARD DEVIATIONFROMRAWDATA 𝒙 = 𝟓. 𝟒𝟕 Distinct Values 𝒙 − 𝒙 𝒙 − 𝒙 𝟐 Frequencies 𝒇 𝒙 − 𝒙 𝟐 1 1 − 5.47 = −4.47 −4.47 2 = 19.9809 𝟏𝟗. 𝟗𝟖𝟎𝟗 2 −3.47 12.0409 𝟎 3 −2.47 6.1009 𝟑𝟎. 𝟓𝟎𝟒𝟓 4 −1.47 2.1609 𝟏𝟓. 𝟏𝟐𝟔𝟑 5 0.47 0.2209 𝟎. 𝟖𝟖𝟑𝟔 6 0.53 0.2809 𝟏. 𝟏𝟐𝟑𝟔 7 1.53 2.3409 𝟒. 𝟔𝟖𝟏𝟖 8 2.53 6.4009 𝟏𝟗. 𝟐𝟎𝟐𝟕 9 3.53 12.4609 𝟐𝟒. 𝟗𝟐𝟏𝟖 10 4.53 20.5209 𝟒𝟏. 𝟎𝟒𝟏𝟖 (𝒙 − 𝒙 ) = 0 𝒙 − 𝒙 𝟐 = 82.509 𝟏𝟓𝟕. 𝟒𝟔𝟕 CALCULATE THE VARIANCE &STANDARD DEVIATION - FREQUENCY TABLE: NUBEROFFRAUDULENT CHEQUESPERWEEK Distinct Values Frequency Squared Observation 1 1 1 2 0 4 3 5 9 4 7 16 5 4 25 6 4 36 7 2 49 8 3 64 9 2 81 10 2 100 VARIANCE & STANDARD DEVIATION FROM UNGROUPED FREQUENCY DATA 𝒔 𝟐 = 𝟏 𝒏 − 𝟏 𝒇𝒙 𝟐 − 𝒇𝒙 𝟐 𝒏 • Variance: 𝒔 𝟐 = 𝟏 𝟑𝟎 − 𝟏 𝟏𝟎𝟓𝟒 − 𝟏𝟔𝟒 𝟐 𝟑𝟎 = 𝟓. 𝟒𝟐𝟗𝟗 • Standard deviation: 𝑠 = 𝜎 = 5.4299 = 𝟐. 𝟑𝟑 FIND THE VARIANCE & STANDARD DEVIATION - GROUPED FREQUENCY TABLE: TruckData: weights(intonnes)of20fullyloadedtrucks Class Intervals Frequency Midpoint Squared Midpoint 𝟐. 𝟓 ≤ 𝒙 ≤ 𝟑. 𝟎 4 2.75 7.5625 𝟑. 𝟎 < 𝒙 ≤ 𝟑. 𝟓 1 3.25 10.5625 𝟑. 𝟓 < 𝒙 ≤ 𝟒. 𝟎 5 3.75 14.0625 𝟒. 𝟎 < 𝒙 ≤ 𝟒. 𝟓 3 4.25 18.0625 𝟒. 𝟓 < 𝒙 ≤ 𝟓. 𝟎 3 4.75 22.5625 𝟓. 𝟎 < 𝒙 ≤ 𝟓. 𝟓 3 5.25 27.5625 𝟓. 𝟓 < 𝒙 ≤ 𝟔. 𝟎 1 5.75 33.0625
  • 9. 2013/05/22 9 VARIANCE & STANDARD DEVIATION FROM GROUPED DATA 𝒔 𝟐 = 𝟏 𝒏 − 𝟏 𝒇𝒙 𝟐 − 𝒇𝒙 𝟐 𝒏 • Variance: 𝒔 𝟐 = 𝟏 𝟐𝟎 − 𝟏 𝟑𝟒𝟖. 𝟕𝟓 − 𝟖𝟏. 𝟓 𝟐 𝟐𝟎 = 𝟎. 𝟖𝟕𝟓𝟕 • Standard deviation: 𝑠 = 𝜎 = 0.8757 = 𝟎. 𝟗𝟒 CALCULATIONS Raw Data IQR Variance & Standard Deviation Frequency Table (Ungrouped Data) IQR Variance & Standard Deviation Frequency Table (Grouped Data) IQR Variance & Standard Deviation BOX - AND - WISKER DIAGRAM (5 POINT SUMMARY) Minimum Value 𝑸 𝟏 = 𝑷 𝟐𝟓 Median𝑸 𝟑 = 𝑷 𝟕𝟓 Maximum Value EXAMPLE Consider the following set of 23 scores: 0 3 4 8 9 12 14 15 16 16 16 18 19 21 22 25 32 34 39 43 54 67 77 1. Find the 5 point summary 2. Draw a box – and – wisher plot to illustrate the values 5 - POINT SUMMARY 0 3 4 8 9 12 14 15 16 16 16 18 19 21 22 25 32 34 39 43 54 67 77 HOMEWORK •Example X-Kit textbook page 218 – 223. •“Practise for your exams” page 224 number 1 & 2. •Par B.2 (page B5) all odd number questions.