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Parameters and Statistics. A statistic is a descriptive measure computed from a sample of data. A parameter is a descriptive measure computed from an entire population of data. Measures of Central Tendency - Arithmetic Mean -. - PowerPoint PPT Presentation
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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.
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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.
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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
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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
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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.
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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.
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Measures of Central Tendency- Mode -
The modemode, , if one exists, is the most frequently occurring observation in the sample or population.
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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.
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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.
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Shapes of the Distribution
Symmetric Distribution
0123456789
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
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Measures of Variability- The Range -
The rangerange is in a set of data is the difference between the largest and smallest observations
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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
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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
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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
)(
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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
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Measures of Variability- Population Standard Deviation-
The population standard deviation, population standard deviation, ,, is
N
xN
ii
1
2
2
)(
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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
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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
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Coefficient of Variation
The population coefficient of variationpopulation coefficient of variation is
0100
ifCV
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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.
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Percentiles and Quartiles
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
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Percentiles and Quartiles
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
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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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