Download Mean &StDeviation

Describing Data with Numbers
Describing Center


Mean: average of all
data
Median: middle
number when in order
Describing Spread or Variability
•Range: Max – Min
•IQR: Q3 – Q1
•Standard deviation(, s) is the average
distance values fall from the mean
Sample
Standard Deviation
 x  x 
2
s
i
n 1
Variance is 2 (or s2) and is
another measure of spread, but in
square units
Computing Standard Deviation

Sample data set: 72, 88, 96, 100
x
xi
x i  x  =
 x i  x 2
72
(72 - 89) = -17
(-17)2 = 289
88
(88 - 89) = -1
(-1)2 = 1
96
(96 - 89) = 7
(7)2 = 49
100
(100 - 89) = 11
(-1)2 = 121
= 89
Sum = 460
460
s
 12.382
4 1
Interpretation: On average, the individual data values are
about 12 units away from the mean.
Calculated Standard Deviation
is a measure of Variation in data
Sample Data Set
Mean
Standard
Deviation
Shape of
Graph
100, 100, 100, 100, 100
100
0
Symmetric
& Uniform
100
10
Symmetric
100
50
Symmetric
142
99.85
Skewed
90, 90, 100, 110, 110
30, 90, 100, 110, 170
90, 90, 100, 110, 320
Which measures of center and
spread do I use when describing a
distribution?


When the sampling distribution is bell shaped
and symmetrically distributed, use the mean
as the measurement for center and standard
deviation as the measure for spread.
When the sampling distribution is unknown or
skewed use the median as the measurement
for center and the IQR &/or range as the
measure for spread.
Using Calculator

Consider the data set:
10, 12, 12, 15, 18
Mean
Median Range St.Dev. Variance
13.4
12
8
3.13
9.79
After adding 5 to 18.4
each data point
17
8
3.13
9.79
After multiplying 40.2
each data point
by 3
36
24
9.39
88.17
Original data
Changing Units

What happens to statistics when all the data is
increased (or decreased) by some number?

Measures of center


Measures of spread


Mean and median are increased by the same number
Range, st dev, and IQR stay the same and do not change
What happens to statistics when all the data is
multiplied by some number?

Measures of center


Mean and median are multiplied by the same number
Measures of spread


Range, st dev, and IQR are multiplied by the same number
Variance is multiplied by the number squared
Changing Units Practice

Suppose a data set has mean 8 and standard
deviation 2.5. What will the new mean and
standard deviation be if:

All data points are multiplied by 4?


All data points are decreased by 2?


Mean = 32; standard deviation = 10
Mean = 6; standard deviation = 2.5
All data points are multiplied by 4 and then decreased
by 2?

Mean = 30; standard deviation = 10