Topics for Today
Measures of Central Tendency Mean, Median, Mode Sample and Population Mean Weighted Means Selecting Appropriate Measures of
Central Tendency Measures of Dispersion
Variance Standard Deviation
Descriptive vs. Inferential
Descriptive Statistics Methods for organizing and
summarizing information
Inferential Statistics Methods for drawing and measuring
the reliability of conclusions about a population based on information obtained from a sample of the population
Looking at This Data Set…
Student Performance in Class Tests
ID Test 1 Test 2 Test 3 Test 4
1 2463 B+ A 95 102 4140 A- A 90 9.53 1210 D F 0 04 O649 D+ B+ 80.5 95 2925 B ? 86 8.56 4194 A- A 86.5 97 4266 B+ F 90.5 8.58 2517 A- A 83.5 10
Overview Mean Median Mode Sample and Population Mean Weighted Means Selecting Appropriate Measures of
Central Tendency Applying these measures
Mean
The mean of a set of n observations is the arithmetic average
Mean of n observations x1, x2,x3,….xn is
In Excel, =AVERAGE(insert range)
ixxn
1
i n
ix
Median
The data value that is exactly in the middle of an ordered list if the number of pieces of data is odd
The mean of the two middle pieces of data in an ordered list if the number of pieces of data is even
The median is a typical value; it is the midpoint of observations when they are arranged in an ascending or descending order
Mode The most frequent data value; i.e., any
value having the highest frequency among the observations
In Excel,you use the functions =MEDIAN (insert range)
=MODE (insert range) Unimodal, Bimodal, Multimodal data
sets Outliers
Sample and Population Means
Mean of a data set Population mean if data set includes
entire population
Sample mean if data set is only a sample of the population
iX
N
ixxn
Weighted Means
To calculate the mean when your information is available only in the form of summary data
C Interval Freq25 – 29.9 430 – 34.9 535 – 39.9 12
j jx fx
n
Skewed Distributions When there is one mode and the distribution
is symmetric mean, median, mode are the same
Positive skew mean moves towards the positive tail median also pulls towards the positive tail
Negative skew mean moves towards the negative tail median also moves towards the negative tail
Selecting Appropriate Measures
Mean affected by extreme values includes all observations, therefore
comprehensive (useful for interval/ratio data) Median
not affected by the number of observations reveals typical situations (used for ordinal data)
Mode useful for nominal variables
Other Useful Calculations
In addition to the sum of data, xwe need to be able to calculate:
2 2; ;x x x x x
2 2x x
xy x y
Variability or Spread Mean and the median - limits Range – coarse measure of variability Percentiles
kth percentile is the point at which k percent of the numbers fall below it and the rest are fall above it
25th percentile (lower quartile) 50th percentile (median) 75th percentile (upper quartile) Interquartile range (difference between the 25th
percentile value and the 75th percentile value)
Describing the Spread
A five number summary Median Quartiles Extremes
Variance and Standard Deviation Measures spread about the mean Standard deviation cannot be discussed
without the mean
Calculating PercentilesIn the list of twelve observations2 4 7 11 11 11 11 14 16 16 24 29Compute median, 25th and 75th percentiles
11 11
2
Median
The lower quartile is the median of the 6 observations that fall below the medianThe upper quartile is the median of the 6 observations that fall above the median
7 112
16 162
Five Number Summary
Median = 11 Lower Quartile = 9 Upper Quartile = 16 Extremes are 2 and 29 Can compute the range = 27 In a symmetric distribution, the lower
and upper quartiles are equally distant from the median
Variance Is the mean of the squares of the
deviations of the observations from their mean
Population variance
Sample variance
2
2
iX
N
2
2
1
ix x
sn
ExampleThe heights, in inches for five starting players in a
men’s college basket ball team are:
67 72 76 76 84
Compute the mean and standard deviation.
x67 -8 6472 -3 976 1 176 1 184 9 81
375 0 156
2x xx x
xx
n= 75
Standard Deviation Standard deviation is positive
square root of the variance
Variance in our basketball example:
2
2
1
ix x
sn
2 156
4s = 39
Formulas – Standard Deviation
2
1
ix x
sn
Standard deviation of a sample
Standard deviation of a population
2
iX
N
Short Cut – Simpler Formula
22
1
n x x
sn n
Standard Deviationof a sample
Sum of the squares of data values, i.e., you square each data value and then sum those squared valuesSquare of the sum of data values, i.e., you sum all the data values and then square that sum
2 x
2
x
Example (using the short cut)
x67 448972 518476 577676 577684 7056
375 28281
2x
2 2
375
140625
x
25 28281 375
5 4
780
20
s
s
39
6.24
s
s
Interpreting Std. Deviation
s and s 2 will be small when all the data are close together
The deviations from the mean Will be both positive and negative Sum will always be 0
s is always 0 or a positive number s = 0 means no spread; as s value
increases, the spread of the data increases The units of s are the same as the original
observations s is heavily influenced by outliers