Section 9.1 Samples and Central Tendency

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.