Statistics
Numerical Descriptive Summary
Summation Notation
Observations in a dataset are denoted by { x1,x2,x3,x4,....xn }; n = sample size x1 is the first observation, x2 is 2nd obs. and so on. We use xi to denote x1 + x2 + x3 + x4 +.+ xn In particular,
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Example 5.0
Suppose a dataset contains obs.
{3.5,11.2,8,.4,5.4}, Then n=5, and
Not Same
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Population and Sample
POPULATION Collection of all the units that we are interested in studying. SAMPLE A subset of the units of the population.
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Numerical Summary
Want to summarize the data using numerical descriptive measures. Two quantities to measure:
I.
CENTER
Measure of central tendency
III.
VARIABILITY
Spread of the data Both population and sample data can be summarized.
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Central Tendency
We want to locate the center of the data in three ways
MEAN MEDIAN MODE
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Mean
MEAN (also called the Arithmetic Mean) is the average of a group of numbers.
Applicable Not
for quantitative data
applicable for qualitative data.
Affected
by each value in the data set, including extreme values. Computed by summing all values in the data set and dividing the sum by the number of observations in the data set.
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Population Mean
The population mean (denoted by ) computes the mean of a population data. Suppose a population contains N obs. {X1,X2,X3,X4,....XN }, Then the population mean
Example: Population: {20,13,18,26,11}, Then N=5, and the population mean
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Sample Mean
The sample mean (called x-bar) computes the mean of a sample data. Suppose a sample contains n obs. {x1,x2,x3,x4,....xn }, Then the sample mean
Example : Suppose a sample contains 5 observations {3.5,11.2,8,.4,5.4}, Then n=5, and sample mean
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Distribution and Mean
Mean the is point where the histogram is balanced. For positively skewed distribution extreme observations will pull it up.
Mean (balanced)
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Median
MEDIAN is the middle most observation.
Middle value in an ordered array of numbers. Not affected by extremely large and extremely small values.
Median partitions the histogram into two equal halves
Mean
50%
Median
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Finding the Median
Arrange the observations in order of magnitude (smallest to largest). If number of observations (n) is odd, the median is the middle term [(n+1)/2 th observations] of the ordered array. If there are even number of observations, the median is the average of the middle two terms [(n/2) and (n/2+1) th observations].
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Example: n is Odd
Suppose a dataset contains the following observations { 3.5,11.2, 8, .4, 5.4 }. We want to find the median. Step I: Rearrange5.4, data 11.2 0.4, 3.5, the 8.0, : Step II: Find the middle value: (n+1)/2th Observation is the (5+1)/2 = 3rd observation. Sample Median=5.4
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Example: n is Even
Suppose a dataset contains the following observations {9.8,2.4,6.2,3.5,11.2,8,.4,5.4}. We want to find the median. Step I : Rearrange the data :
0.4, 2.4, 3.5, 5.4, 6.2, 8.0, 9.8, 11.2 Step II : Find two middle terms. Middle observation are the 8/2 = 4th and the 5th observations. Sample Median = (5.4+6.2)/2 = 5.8
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Mode
MODE is the most frequent observation. For continuous variables, mode is the point where the histogram has the peak.
he most frequently occurring value in the data T
set
ata displayed in a histogram will have a modal D
class the class with the largest frequency
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Example
The Data set 1 3 5 6 8 8 9 11 12
Mean Median is the Mode is 8
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or 5th observation, 8