section 9.1 samples and central tendency
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Section 9.1 Samples and Central Tendency. Objectives: 1.To distinguish population parameters from sample statistics. 2.To find measures of central tendency. - PowerPoint PPT PresentationTRANSCRIPT
Section 9.1
Samples and Central Tendency
Section 9.1
Samples and Central Tendency
Objectives:1. To distinguish population
parameters from sample statistics.
2. To find measures of centraltendency.
Objectives:1. To distinguish population
parameters from sample statistics.
2. To find measures of centraltendency.
Researchers use a small group of people, called a sample, to
approximate information about a larger group, called the
population.
Researchers use a small group of people, called a sample, to
approximate information about a larger group, called the
population.
AA populationpopulation is the complete is the complete collection of elements (scores, collection of elements (scores, people, measurements) to be people, measurements) to be studied.studied.
A A samplesample is a subset of a is a subset of a population.population.
DefinitionDefinitionDefinitionDefinition
The larger group of interest is called the population, and the selected subset is a sample.
The larger group of interest is called the population, and the selected subset is a sample.
If every member of the population has an equal
chance of being included in the sample, then it is a random
sample.
If every member of the population has an equal
chance of being included in the sample, then it is a random
sample.
A stratified random sample is a random sample within certain
groups of a population.
A stratified random sample is a random sample within certain
groups of a population.
A stratified random sample is a sample obtained by separating the population elements into
nonoverlapping groups, called strata, and then selecting a
simple random sample within each stratum.
A stratified random sample is a sample obtained by separating the population elements into
nonoverlapping groups, called strata, and then selecting a
simple random sample within each stratum.
Parameter Parameter The actual value of The actual value of a quantity for the population, a quantity for the population, usually known only to God.usually known only to God.
**Usually represented with Usually represented with Greek lettersGreek letters
DefinitionDefinitionDefinitionDefinition
Statistic Statistic An estimate of the An estimate of the population parameter based on population parameter based on a sample.a sample.
**Usually represented with Usually represented with English lettersEnglish letters
DefinitionDefinitionDefinitionDefinition
Greek letters usually represent parameters, while English
letters represent statistics. For example, μ represents the
population mean while x (x bar) represents the sample mean.
The bar over the x distinguishes the sample mean from an individual value of the
variable x.
Greek letters usually represent parameters, while English
letters represent statistics. For example, μ represents the
population mean while x (x bar) represents the sample mean.
The bar over the x distinguishes the sample mean from an individual value of the
variable x.
The number (n) of values is the sample size, the number in the population is N, and the data
values are numbered with subscripts x1, x2, x3, . . . xn.
These values can be referred to as xi for i = 1, 2, 3, . . . n.
The number (n) of values is the sample size, the number in the population is N, and the data
values are numbered with subscripts x1, x2, x3, . . . xn.
These values can be referred to as xi for i = 1, 2, 3, . . . n.
The letter i is a counter variable or index. The symbol
is used to represent the addition of data values
because it is the capital Greek letter sigma that corresponds
to our letter s as an abbreviation for sum.
The letter i is a counter variable or index. The symbol
is used to represent the addition of data values
because it is the capital Greek letter sigma that corresponds
to our letter s as an abbreviation for sum.
The starting value of the index appears below the and the
ending value above the . The summation in the following
definition is read “summation of x sub i as i goes from 1 to
n.”
The starting value of the index appears below the and the
ending value above the . The summation in the following
definition is read “summation of x sub i as i goes from 1 to
n.”
MeanMean
where where nn is the sample size is the sample sizenn
xxxx
nn
i = 1i = 1ii
==
DefinitionDefinitionDefinitionDefinition
The mean is one of several statistics that are called
measures of central tendency.
The mean is one of several statistics that are called
measures of central tendency.
Median Median The middle value (or The middle value (or average of the middle two average of the middle two values) after listing the data in values) after listing the data in order of size.order of size.
DefinitionDefinitionDefinitionDefinition
Mode Mode The most frequent The most frequent value(s) (if any).value(s) (if any).
DefinitionDefinitionDefinitionDefinition
Midrange Midrange The average of the The average of the highest and lowest value.highest and lowest value.
DefinitionDefinitionDefinitionDefinition
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88 x1, x2, x3, x4, x5, x6, x7, x8x1, x2, x3, x4, x5, x6, x7, x8
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the sample size.
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the sample size.
n = 8n = 8
= 85 + 93 + 96 + 74 + 65 = 85 + 93 + 96 + 74 + 65
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the mean.
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the mean.
==
88
11iiiixx
= 676= 676
+ 88 + 87 + 88+ 88 + 87 + 88
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the mean.
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the mean.
x =x =nn
xixi
nn
i =1i =1= 84.5= 84.5==
6768
6768
55..8787==22
88888787 ++
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the median.
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the median.
65, 74, 85, 87, 88, 88, 93, 9665, 74, 85, 87, 88, 88, 93, 96
65, 74, 85, 87, 88, 88, 93, 9665, 74, 85, 87, 88, 88, 93, 96
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the mode.
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the mode.
55..8080==22
96966565 ++
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the midrange.
Practice: The following test scores were recorded:
85, 93, 96, 74, 65, 88, 87, 88
Find the midrange.
65, 74, 85, 87, 88, 88, 93, 9665, 74, 85, 87, 88, 88, 93, 96
nn
i =1i =1
nn
i =1i =1
nn
i =1i =1(xi + yi) = xi + yi(xi + yi) = xi + yi
nn
i =1i =1k = nk for any k R k = nk for any k R
Summation Rules: Summation Rules:
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
sample size
n = 8
sample size
n = 8
meanmean
= 685= 685
i =1i =1xi = 76 + 86 + 78 + 83 + 90 +xi = 76 + 86 + 78 + 83 + 90 +
88
88 + 94 + 9088 + 94 + 90
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
meanmean
88685685
x =x = = 85.625 ≈ 85.6= 85.625 ≈ 85.6==nn
nn
i =1i =1xixi
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
median
76, 78, 83, 86, 88, 90, 90, 94
median
76, 78, 83, 86, 88, 90, 90, 94
= 87= 8722
86 + 8886 + 88
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
mode
76, 78, 83, 86, 88, 90, 90, 94
mode
76, 78, 83, 86, 88, 90, 90, 94
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
midrange
76, 78, 83, 86, 88, 90, 90, 94
midrange
76, 78, 83, 86, 88, 90, 90, 94
= 85= 8522
76 + 9476 + 94
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90.
Homework:
pp. 451-453
Homework:
pp. 451-453
■ Cumulative Review
27. Solve ABC if b = 29, c = 21, and C = 42°.
■ Cumulative Review
27. Solve ABC if b = 29, c = 21, and C = 42°.
■ Cumulative Review
28. Find the central angle of a circle of radius 10.2 m if the angle
intercepts an arc of length 47.9 m. Give the answer in both radians and degrees.
■ Cumulative Review
28. Find the central angle of a circle of radius 10.2 m if the angle
intercepts an arc of length 47.9 m. Give the answer in both radians and degrees.
■ Cumulative Review
29. Which elementary row operation changes the sign of the
determinant?
■ Cumulative Review
29. Which elementary row operation changes the sign of the
determinant?
■ Cumulative Review
30. Find 5<2, 7> – 2<1, -6>
■ Cumulative Review
30. Find 5<2, 7> – 2<1, -6>
■ Cumulative Review
31. Write the equation of an ellipse with center (5, -2), horizontal major axis of 10, and eccentricity of 0.75.
■ Cumulative Review
31. Write the equation of an ellipse with center (5, -2), horizontal major axis of 10, and eccentricity of 0.75.