统计学外文翻译

外文翻译原文

名称：

Fundamentals_of_Statistics

Measures of Central Tendency and Location:

mean, median, mode, percentiles, quartiles and deciles.

x

sorted x

53

53

55

53

70

53

58

55

64

57

57

57

53

58

69

64

57

68

68

69

53

70

The Measures of Central Tendency are Mean, Median and Mode

Mean

?

x-bar

or

x

?

for

a

given

variable,

it

is

the

sum

of

the

values

divided

by

the

number

of

values (

?

x

i

/< /p>

n

).

In this case, we have

n

= 11.

So we need to add all of the values together and

divide by 11.

?

= 657,

x

= 59.73

Median

?

the number in a distribution of a variable’s response where one half of the values are above and

one half of the values are below.

To find the median, we first need to put our data in ascending

order

(smallest

to

largest).

Then

we

can

determine

the

median…if

the

value

of

n

is

odd,

it

is

simply the middle observation, but if the value of

n

is even, it is the average of the two middle

observations.

In this case,

n

is odd, so the median will be the middle observation of our sorted values (the 6

th

value)...57

Mode

?

the

value

that

occurs

most

frequently.

If

there

are

two

different

values

most

frequently

occurring, the data are said to be bi- modal.

If there are more than two modes, and the distribution is

said to be multi-modal.

In this case, the value that occurs most often is 53.

So, the mode is 53.

The measures of location are Percentile, Quartile and Decile

Percentile

?

the p

th

percentile is a value such that

at least p

percent of the observations are less than

or equal to this value

and

at least (100

–

p)

percent of the observations are greater than or

equal to this value.

To calculate percentiles, we use indices (

i

).

i =

(p/100) n

for p

1

, p

2

, p

3

,…p

99

If the answer is a whole number (an integer), then

i

is the average of

(P/100)n

and

1

+ (P/100)n

.

If the index number is not a whole number, we ALWAYS round up.

The position

of the index is the next whole number (integer) greater than the computed index.

For example:

i

(p50)

= (50/100)11 = 5.5...this rounds up to 6

So, we would count from the lowest value of the sorted data to the index number (6).

Since the calculated

i

was not a whole number we had to round up to find the value

where at least 50% of the values are equal to or lower than this value and at least 50%

are equal to or higher than this value.

In this case, the value of the 50

th

percentile

is the 6

th

value..

.57 … Does this look familiar?

?

The 50

th

percentile is the same

thing as the median.

What does it tell us?

In this distribution, AT LEAST 50% of the observations are

LESS THAN OR EQUAL TO 57 AND AT LEAST 50% of the observations are

GREATER THAN OR EQUAL TO 57.

i

(p80)

= (80/100)11 = 8.8...this round up to 9.

The 9

th

value is 68.

Again, since the index number is not a whole number, we round up. So, we would count from the

lowest value of the sorted data to the index number (9).

In this case, the value of the

80

th

percentile is 68.

Since this dataset has 11 observations, we won’t have any instances where our calculated index

number is a whole number.

However, if we just remove our value of 70 and create a

new distribution, we will be able to see an example...

53 53 53 55 57 57 58 64 68 69

i

(p30)

= (30/100)10 = 3...this is a whole number, so we must take the 3

rd

and 4

th

values and

average them to find the 30

th

percentile.

(53 + 55)/2 = 54

So, the value of the 30

th

percentile is 54.

Return to our original data distribution ...

Quartiles

–

are special cases of percentiles…Q

1

= P

25

, Q

2

= P

50

, Q

3

= P

75

,

These three values divide the distribution into 4 equal quarters

i

(Q1)

= (25/100)11 =

2.75...this rounds to 3, so Q1 is the 3

rd

value...53

i

(Q2)

= (50/100)11 = 5.5...this round to 6, so Q2 is the 6

th

value...57

i

(Q3)

= (75/100)11 = 8.25...this rounds to 9, so Q3 is the 9

th

value...64

Measures of Dispersion or V

ariability

:

Range, interquartile range (IQR), variance, standard deviation

and coefficient of variation.

Range

= This tells us how wide the span is from the maximum value to the minimum value.

(Max

–

Min) = Range.

In this instance, the range is 69 - 53 = 16.

Interquartile Range (IQR)

= This tells us how wide the span is in the middle 50% of the data.

(Q3

–

Q1) = IQR.

In this case ... 64

–

53 = 11

We will use IQR in later processes, so we will want to keep this

x

(x-xbar)

(x-xbar)

2

53

-6.73

45.29

53

-6.73

45.29

53

-6.73

45.29

55

-4.73

22.37

57

-2.73

7.45

57

-2.73

7.45

58

-1.73

2.99

64

68

69

70

657

657/11=59.73

4.27

8.27

9.27

10.27

-0.03

18.23

68.39

85.93

105.47

454.18

454.18/10

≈

45.2

?

(

x

?

x

)

We use the formula:

n

?

1

2

=

s

2

The variance for these data is 454.18.

For our purposes here, the computation of variance

is just a step towards the computation of the standard deviation.

Sample standard deviation (

s

)

is the positive square root of the variance.

?

45< /p>

.

42

?

6

.

74

= s

So the formula for sample standard deviation is…

?< /p>

(

x

?

x

)

n

?

1

Population

Variance

< br>(

?

2

)

?

uses

the

same

formula

in

the

numerator,

but

N

instead

of

n-1

in

the

denominator.

Since

we

rarely

have

information

about

the

entire

population,

we

almost

always use the formula for sample variance,

s

2

.

Population

Standard

Deviation:

?

=

?

2

…since

we

rarely

have

information

from

the

entire

population, we use the formula for sample standard deviation,

s

.

Coefficient of Variation:

?

?

100

tells us what percent the sample standard deviation is of

the sample mean

This number

is “relative” and is only of use in comparing the distribution of two or more

variables.

Suppose I have two samples, and I want to know which sample has more variability…

If

both

samples

have

the

same

mean,

the

one

with

the

higher

standard

deviation

will

have the greater variability.

However, if they have different means, I need to calculate

the coefficient of variation to determine which one has the most variability.

xbar = 458,

s = 112 versus xbar = 687, s = 192

Standardized Data and Detecting Outliers

Z

-score:

z

=

?

s< /p>

?

?

x

?

x

?

x

s

The z-score tells us how many standard deviations a value is from the mean.

We can look at a

picture of what a z-score tells u

s.

In the Normal Curve…the mean is at the highest point and the

curve tails off symmetrically in both directions.

The sign of the z-score tells us which direction the value is from the mean on the Normal Curve.

Negative values will be to the left, and positive values will be to the right.

Standardizing Scores:

Standard

Normal

Curve

…the

mean

is

zero,

and

the

standard

deviation

is

1.

The

distribution

is

bell-shaped

and

symmetrical.

The area

under

the curve

is

1,

and

the

tails

of

the

curve

extend

out

infinitely.

They never actually touch the horizontal axis.

The highest point on the curve is at the

mean

Return to our data …let’s calculate the z

-

scores for each of the values…

Empirical Rule

?

used when the distribution is assumed to known to be approximately

normal.

?

Approximately 68% of the values will fall within 1

sd

of the mean

?

Approximately 95% of the values will fall within 2

sd

of the mean

?

Approximately 99.9% of the values will fall within 3

sd

of the mean

Chebyshev’s Theorem

?

doesn’t require that the data have a normal distribution

Says that at least (

1

–

1/z

2

) values will fall within z standard deviations of the mean.

1-1/1

2

= 0,

1-1/2

2

= .75,

1-1/3

2

= .88889,

1-1/4

2

= .9375,

1-1/5

2

= .96

?

We can’t make a

ny assumptions about the percent of values that are within 1

sd

of the

mean

But…

?

At least 75% of the values will fall within 2

sd

of the mean

?

At least 88.9% of the values will fall within 3

sd

of the mean

We use Chebyshev’s

Theorem to estimate the variation in a distribution when

?

n

< 30, or

?

the shape of the distribution is unknown, or

?

the distribution is assumed to be non-normal.

Outliers:

suspect or extreme values of data that must be identified and scrutinized.

If they

are instances of incorrectly entered data, they should be corrected.

If the value was

entered correctly and it is a valid number, it should remain in the dataset as part of the

initial analysis.

When we use the z-score method for identifying outliers, we assume that any value that has a z-score

with

an

absolute

value

greater

than

3.0

(that

is

less

than

-3.0

or

greater

than

+3.0)

is

an

outlier.

Before we proceed with data analysis, we need to examine all outliers for accuracy.

If we determine

that the value is valid, we often run two sets of analysis.

One with the outlier, and one without.

Another way to identify outliers…

Related to IQR is the Five number summary

…minimum, Q1, Q2, Q3, & maximum.

These

values feed into upper and lower limits, and we graph them in a box plot.

Five Number Summary

Minimum

53