Transcript mean - Lake Travis ISD

Symbols Used to Describe Data
• Parameters—summary
measurements of the
population data
• : population mean
• p: population
proportion
• : population standard
deviation
• Statistics—summary
measurements of the sample
data.
• x : sample mean
• p̂ : sample proportion
• s : sample standard
deviation
Measures of Center
Mean(, x ) —add up data values and divide by
number of data values
Median (M)—list data values in order, locate
middle data value; average middle 2 if necessary
Data Set: 19, 20, 20, 21, 22
Mean = 20.4; Median = 20
Data Set: 19, 20, 20, 21, 38
Mean = 23.6; Median = 20
Robust (Resistant) Statistic
• Robust or resistant: value doesn’t change
dramatically when extreme values, like outliers,
are added to (or taken out of) the data set.
– Median is resistant, so it is used to state the “typical”
value
– Mean is NOT resistant against extreme values. Mean
is pulled away from the center of the distribution
toward the extreme value (“tails of graph”).
Mean or
Median?
Measures of Center on
Different Distribution Shapes
In each of the graphs, decide which mark
represents the mean µ, median M, and mode Mo.
µ, M, Mo
Skewed to the left
all =
Symmetric
Mo, M, µ
Skewed to the right
Remember the mean is pulled toward extreme values.
Describing Spread: Range, IQR,
Standard Deviation, & Variance
•Range: Max – Min
•IQR: Q3 – Q1
•Standard deviation(, s): the average
distance values fall from the mean
•Variance: standard deviation squared
Population and Sample
Standard Deviation
 x   
2

i
n
 x  x 
2
s
i
n 1
Variance is 2 (or s2) and is another measure
of spread, but in square units
If data is in your calculator, press Stat, Calc,
1-Var stats and the Sx is the standard deviation
Computing & Interpreting
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
(11)2 = 121
= 89
Sum = 460
460
s
 12.382
3
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
100, 100, 100, 100, 100
Mean
Standard Shape of
Deviation Graph
100
0
Symmetric,
(One bar)
90, 90, 100, 110, 110
100
10
Symmetric
30, 90, 100, 110, 170
100
50
Symmetric
90, 90, 100, 110, 320
142
99.85
Skewed
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.
Reading Computer Output
• What type of graph could be made with the given
information?
• Boxplot using the 5 number summary
• Determine if there are any outliers or not.
• 1.5(IQR) = 39.99;
• 19.06 - 39.99 = -20.93; 45.72 + 39.99 = 85.71
• Outliers would be < -20.94 or >85.71
• Since max is 93.34, it is an outlier
• Not known where right whisker would end, but
somewhere between Q3 and 85.71
How are measures of center and
spread effected by conversions?
• Suppose we have a data set: 2,5,7,8,9
• If we add 5 to each number in the data set,
– what will happen to the mean and median?
• They will increase by 5, since everything is shifted up
– what will happen to the range/IQR/st dev.?
• They will stay the same, since the spread of the data
remains the same
Data set:2,5,7,8,9
• If we multiply each number by 2,
– What will happen to the mean and median?
• They will also multiply by 2, because they all
double
– What will happen to the range/IQR/st dev.?
• They will also multiply by 2, since the spread gets
magnified as well.
– What will happen to the variance?
• It will be multiplied by 22, or 4, since the variance is
a squared measure.
When data is converted:
• Measures of center are effected by all
operations
– Do to the mean and median what you do to the
data
• Measures of spread are only effected by
multiplication (and division).
– Addition/subtraction does not change spread!
Example
The mean height of a class of 15 children is 48 inches, the
median is 45 inches, the standard deviation is 2.4 inches, and the
IQR is 3 inches. Find the mean, median, standard deviation, and
IQR if:
– you convert each height to feet
• Divide data by 12, so divide everything by 12
• Mean = 4, median = 3.75, sd = 0.2, IQR = 0.25 (all in feet)
– each child grows 2 inches
• Add 2 to data, so add 2 to measures of center only
• Mean = 50, median = 47, sd = 2.4, IQR = 3 (all in inches)
– each child grows 2 inches and you convert the height to
feet
• Add 2 then divide by 12 to data, so add 2 to measures of center
then divide everything by 12
• Mean = 4.17, median = 3.92, sd = 0.2, IQR = 0.25 (all in feet)