summary statistics

5/3/2023 Summary Statistics 1
Summary Statistics
Last week we used stemplots and histograms to
describe the shape, location, and spread of a
distribution. This week we use numerical summaries of
location and spread.

Main Summary Statistics by Type
Central location
 Mean
 Median
 Mode
Spread
 Variance and standard deviation
 Quartiles and Inter Quartile Range (IQR)
Shape
 Statistical measures of spread (e.g., skewness and
kurtosis) are available but are seldom used in
practice (not covered)

Notation
n  sample size
X  variable
xi  value of individual i
  sum all values (capital sigma)
Illustrative example (sample.sav), data:
21 42 5 11 30 50 28 27 24 52
 n = 10
 X = age
 x1= 21, x2= 42, …, x10= 52
 x = 21 + 42 + … + 52 = 290

Sample Mean

 
 i
i
x
n
n
x
x
1
0
.
29
)
290
(
10
1
1


  i
x
n
x
Illustrative example: n = 10 (data & intermediate calculations on prior slide)

Population Mean
Same operation as sample mean, but
based on entire population (N =
population size)
Not available in practice, but important
conceptually

 
 i
i
x
N
N
x 1


Interpretation of xbar
Sample mean used to predict
 an observation drawn at random from a sample
 an observation drawn at random from the
population
 the population mean
Gravitational center (balance point)
0 10 20 30 40 50 60
Mean = 29

Median – a different kind of average
“Middle value”
Covered last week
 Order data
 Depth of median is (n+1) / 2
 When n is odd  middle value
 When n is even  average two middle values
Illustrative example, n = 10  median has
depth (10+1) / 2 = 5.5
05 11 21 24 27 28 30 42 50 52

median = average of 27 and 28 = 27.5

Median is “robust”
Robust  resistant to skews and outliers
This data set has a mean (xbar) of 1600:
1362 1439 1460 1614 1666 1792 1867
This data set has an outlier and a mean of 2743:
1362 1439 1460 1614 1666 1792 9867
Outlier
The median is 1614 in both instances.
The median was not influenced by the outlier.

Mode
Mode  value with greatest frequency
e.g., {4, 7, 7, 7, 8, 8, 9} has mode = 7
Used only in very large data sets

Mean, Median, Mode
(A) Symmetrical data: mean = median
(B) positive skew: mean > median [mean gets “pulled” by tail]
(C) negative skew: mean < median
Mean Mode
Median
(A)Symmetrica
l
Mode
Median
Mean
Mean
Median
Mode
(B)PositiveSkew (B)NegativeS
kew

Spread = Variability
Variability  amount values spread
above and below the average
Measures of spread
 Range and inter-quartile range
 Standard deviation and variance (this week)

Range = max – min
The range is rarely used in practice b/c it
tends to underestimate population range
and is not robust

Standard deviation
x
xi 
Deviation =
 2
 
 x
x
SS i
Sum of squared deviations =
1
2


n
SS
s
Sample variance =
2
s
s 
Sample standard deviation =
Most common descriptive measure of spread

Standard deviation (formula)
 

 2
)
(
1
1
x
x
n
s i
Sample standard deviation s is the unbiased estimator of
population standard deviation .
Population standard deviation  is rarely known in practice.

New data set (“Metabolic Rates”)
This example is not in your lecture notes
Metabolic rates (cal/day), n = 7
1792 1666 1362 1614 1460 1867 1439
1600
7
200
,
11
7
1439
1867
1460
1614
1362
1666
1792









x

Metabolic rates showing mean (*) and
deviations of first two observations

Standard Deviation Calculation
metabolic.sav – introduced slide 15
Observations Deviations Squared deviations
1792 1792 1600 = 192 (192)2 = 36,864
1666 1666 1600 = 66 (66)2 = 4,356
1362 1362 1600 = -238 (-238)2 = 56,644
1614 1614 1600 = 14 (14)2 = 196
1460 1460 1600 = -140 (-140)2 = 19,600
1867 1867 1600 = 267 (267)2 = 71,289
1439 1439 1600 = -161 (-161)2 = 25,921
SUMS  0* SS = 214,870
x
xi 
i
x  2
x
xi 
* Sum of deviations will always equal zero

Standard Deviation Metabolic data
(cont.)
2
2
calories
67
.
811
,
35
1
7
870
,
214
1





n
SS
s
calories
24
.
189
67
.
811
,
35
2


 s
s
Variance (s2)
Standard deviation (s)

General rule for rounding means
and standard deviations
Report mean to one additional decimals above that of
the data
To achieve accuracy, intermediate calculations should
carry still an additional decimals
Illustrative example
 Suppose data is recorded with one decimal accuracy (i.e.,
xx.x)
 Report mean with two decimal accuracy (i.e., xx.xx)
 Carry all intermediate calculations with at least three decimal
accuracy (i.e., xx.xxx)
Even more important: Always use common sense and judgment.

TI-30XIIS – about $12
In practice, we often use software
or a calculator to check our
standard deviation

Interpretation of Standard Deviation
Larger standard deviation  greater variability
 s1 = 15 and s2 = 10  group 1 has more variability
68-95-99.7 rule – Normal data only
 68% of data with 1 SD of mean, 95% within 2 SD from
mean, and 99.7% within 3 SD of mean
 e.g., if mean = 30 and SD = 10, then 95% of individuals are
in the range 30 ± (2)(10) = 30 ± 20 = (10 to 50)
Chebychev’s rule – All data
 at least 75% data within 2 SD of mean
 e.g., mean = 30 and SD = 10, then at least 75% of
individuals in range 30 ± (2)(10) = (10 to 50)

Quartiles and IQR
Quartiles divide the ordered data into
four equally-sized groups
Q0 = minimum
Q1 = 25th %ile
Q2 = 50th %ile (Median)
Q3 = 75th %ile
Q4 = maximum

Rule for quartiles
Find the median  Q2
Middle of lower half of data set  Q1
Middle of upper half of the data  Q3
Bottom half | Top half
05 11 21 24 27 | 28 30 42 50 52
  
Q1 Q2 Q3
IQR = Q3 – Q1 = 42 – 21 = 21
gives spread of middle 50% of the data

5-Point Summary (sample.sav)
Q0 = 5 (minimum)
Q1 = 21 (lower hinge)
Q2 = 27.5 (median)
Q3 = 42 (upper hinge)
Q4 = 52 (maximum)
Best descriptive statistics for skewed data

Illustrative example (metabolic.sav)
1362 1439 1460 1614 1666 1792 1867

median
Bottom half : 1362 1439 1460 1614

Q1 = (1439 + 1460) / 2 = 1449.5
Top half: 1614 1666 1792 1867

Q3 = (1666 + 1792) / 2 = 1729
5-point summary: 1362, 1449.5, 1614, 1729, 1867

Box-and-whiskers plot (boxplot)
5 point summary + “outside values”
Procedure
 Determine 5-point summary
 Draw box from Q1 to Q3
 Draw line @ Q2
 Calculate IQR = Q3 – Q1
 Calculate fences
 FLower = Q1 – 1.5(IQR)
 FUpper = Q3 + 1.5(IQR)
 Determine if any outside values? If so, plot separately
 Determine inside values and draw whiskers from box to
inside values

Boxplot example
5-point: 5, 21, 27.5, 42, 52
IQR = 42 – 21 = 21
FU = 42 + (1.5)(21) = 73.5
 No outside above (outside)
Upper inside value = 52
FL = 21 – (1.5)(21) = –10.5
 No values below (outside)
 Lower inside value = 5
05 11 21 24 27 28 30 42 50 52
60
50
40
30
20
10
0
Upper inside = 52
Q3 = 42
Q1 = 21
Lower inside = 5
Q2 = 27.5

Boxplot example 2
5-point: 3, 22, 25.5, 29, 51
IQR = 29 – 22 = 7
FU = 29 + (1.5)(7) = 39.5
 One outside (51)
 Inside value = 31
FL = 22 – (1.5)(7) = 11.5
 One outside (3)
 Inside value = 21
3 21 22 24 25 26 28 29 31 51
60
50
40
30
20
10
0
Outside value (51)
Outside value (3)
Inside value (21)
Upper hinge (29)
Lower hinge (22)
Median (25.5)
Inside value (31)

Boxplot example 3 (metabolic.sav)
5-point: 1362, 1449.5, 1614, 1729,
1867 (slide 30)
IQR = 1729 – 1449.5 = 279.5
FU = 1729 + (1.5)(279.5) =
2148.25
 None outside
 Upper inside = 1867
FL = 1449.5 – (1.5)(279.5) =
1030.25
 None outside
 Lower inside = 1362
1362 1439 1460 1614 1666 1792 1867
7
N =
Data source: Moore,
2000
1900
1800
1700
1600
1500
1400
1300

Interpretation of boxplots
Location
 Position of median
 Position of box
Spread
 Hinge-spread (box length) = IQR
 Whisker-to-whisker spread (range or range minus
the outside values)
Shape
 Symmetry of box
 Size of whiskers
 Outside values (potential outliers)

Side-by-side boxplots
Boxplots are especially useful for comparing groups:

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