b a c kn e x t h o m e parameters and statistics statistic a statistic is a descriptive measure...

b a c k n e x th o m e

Parameters and Statistics

A statisticstatistic is a descriptive measure computed from a sample of data.

A parameterparameter is a descriptive measure computed from an entire population of data.


Measures of Central Tendency- Arithmetic Mean -

The arithmetic meanarithmetic mean of a set of data is the sum of the data values divided by the number of observations.


Sample Mean

If the data set is from a sample, then the sample mean, , is:XX

n

xxx

n

xX n

n

ii

211

n

xxx

n

xX n

n

ii

211


Population Mean

If the data set is from a population, then the population mean, , is:

N

xxx

N

xn

N

ii

211

N

xxx

N

xn

N

ii

211


Measures of Central Tendency- Median -

An ordered arrayordered array is an arrangement of data in either ascending or descending order. Once the data are arranged in ascending order, the medianmedian is the value such that 50% of the observations are smaller and 50% of the observations are larger.


Measures of Central Tendency- Median -

If the sample size n is an odd number, the median, Xm, is the middle observation. If the sample size n is an even number, the medianmedian, Xm, is the average of the two middle observations. The medianmedian will be located in the 0.50(n+1)th ordered position0.50(n+1)th ordered position.


Measures of Central Tendency- Mode -

The modemode, , if one exists, is the most frequently occurring observation in the sample or population.


Shape of the Distribution

The shape of the distribution is said to be symmetricsymmetric if the observations are balanced, or evenly distributed, about the mean. In a symmetric distribution the mean and median are equal.


Shape of the Distribution

A distribution is skewedskewed if the observations are not symmetrically distributed above and below the mean. A positively skewedpositively skewed (or skewed to the right) distribution has a tail that extends to the right in the direction of positive values. A negatively skewednegatively skewed (or skewed to the left) distribution has a tail that extends to the left in the direction of negative values.


Shapes of the Distribution

Symmetric Distribution

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10

1 2 3 4 5 6 7 8 9

Fre

qu

ency

Positively Skewed Distribution

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9

Fre

qu

ency

Negatively Skewed Distribution

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9

Fre

qu

ency


Measures of Variability- The Range -

The rangerange is in a set of data is the difference between the largest and smallest observations


Measures of Variability- Sample Variance -

The sample variance, ssample variance, s22,, is the sum of the squared differences between each observation and the sample mean divided by the sample size minus 1.

1

)(1

2

2

n

Xxs

n

ii


Measures of Variability

- Short-cut Formulas for ss22

Short-cut formulas for the sample variance, ssample variance, s22,, are:

11

)(22

21

2

2

n

Xnxsor

nn

xx

s i

n

i

ii


Measures of Variability- Population Variance -

The population variance, population variance, 22, , is the sum of the squared differences between each observation and the population mean divided by the population size, N.

N

xN

ii

1

2

2

)(


Measures of Variability- Sample Standard Deviation -

The sample standard deviation, s,sample standard deviation, s, is the positive square root of the variance, and is defined as:

1

)(1

2

2

n

Xxss

n

ii


Measures of Variability- Population Standard Deviation-

The population standard deviation, population standard deviation, ,, is

N

xN

ii

1

2

2

)(


The Empirical Rule(the 68%, 95%, or almost all rule)

For a set of data with a mound-shaped histogram, the Empirical RuleEmpirical Rule is:

• approximately 68%68% of the observations are contained with a distance of one standard deviation around the mean; 1

• approximately 95%95% of the observations are contained with a distance of 2 standard deviations around the mean; 2

• almost all of the observations are contained with a distance of three standard deviation around the mean; 3


Coefficient of Variation

The Coefficient of Variation, CV,Coefficient of Variation, CV, is a measure of relative dispersion that expresses the standard deviation as a percentage of the mean (provided the mean is positive).

The sample coefficient of variationsample coefficient of variation is

0100 XifX

sCV


Coefficient of Variation

The population coefficient of variationpopulation coefficient of variation is

0100

ifCV


Percentiles and Quartiles

Data must first be in ascending order. PercentilesPercentiles separate large ordered data sets into 100ths. The PPth th percentilepercentile is a number such that P percent of all the observations are at or below that number.

QuartilesQuartiles are descriptive measures that separate large ordered data sets into four quarters.



The first quartile, Qfirst quartile, Q11, is another name for the

2525thth percentile percentile. The first quartile divides the ordered data such that 25% of the observations are at or below this value. Q1 is located in the .25(n+1)st position when the data is in ascending order. That is,

position ordered 4

)1(1

n

Q



The third quartile, Qthird quartile, Q33, is another name for the

7575thth percentile percentile. The first quartile divides the ordered data such that 75% of the observations are at or below this value. Q3 is located in the .75(n+1)st position when the data is in ascending order. That is,

position ordered 4

)1(33

nQ


Interquartile Range

The Interquartile Range (IQR)Interquartile Range (IQR) measures the spread in the middle 50% of the data; that is the difference between the observations at the 25th and the 75th percentiles:

13 QQIQR

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