Download Mean > Median > Mode

5-Minute Check on Activity 7-3
1. What types of graphs were discussed in the previous lesson?
Dot plots, histograms and stem-and-leaf
2. What types of graphs can our calculator do?
Dot and line plots, histograms and box-plots
3. What two types of data are graphed?
Categorical (qualitative) and Quantitative
4. What are the disadvantages of stem-and-leaf plots?
Don’t work well for large amounts of data and not in calculator
5. What is an advantage of back-to-back stem-and-leaf plots?
Allows dataset comparisons – especially shapes
Click the mouse button or press the Space Bar to display the answers.
Activity 7 - 4
Class Survey Continued
Objectives
• Determine measures of central tendency, including
the mean, median, mode and midrange
• Recognize symmetric and skewed frequency
distributions
• Distinguish between percentiles and quartiles
Vocabulary
• Central Tendency – a statistic that measures the
“center” of a distribution
• Mean – the average value
• Median – the middle value (in an ordered list)
• Midrange – the average of the largest and smallest
observations
• Mode – the most frequent data value
• Resistant measure – a measure (statistic or
parameter) that is not sensitive to the influence of
extreme observations
Activity
In the previous activity the following data could have been
representative of a class of 20 students.
Family sizes: 4, 6, 2, 8, 3, 5, 6, 4, 7, 2, 5, 6, 4, 6, 9, 4, 7, 5, 6, 3
The frequency distribution of the data was displayed
graphically using a dotplot and a histogram.
Where does the center of the distribution appear to be?
Between 5 and 6
Activity cont
In the previous activity the following data could have been
representative of a class of 20 students.
Family sizes: 4, 6, 2, 8, 3, 5, 6, 4, 7, 2, 5, 6, 4, 6, 9, 4, 7, 5, 6, 3
What is the mean, median and mode of the distribution?
Mean: average
sum / 20 = 102 / 20 = 5.1
Median is the average of 10th and 11th ordered data:
(5 + 5) / 2 = 5
Mode is the most frequent: 6 (occurs 5 times)
Measures of the “Center”
• Common measures of a center of a distribution are:
– Mean (the average of all the numbers)
– Median (the middle value of all the numbers)
– Mode (the most frequent occurring of the
numbers)
– Midrange (the average of the highest and lowest
number in the distribution)
• The first three are used a lot in statistical analysis
Mean
• Mean: The “average” value of a dataset
x1  x2  ... xn
x
n
x

x
i
n
• What is the mean of the following numbers:
1, 2, 2, 5, 8, 9, 99
Add the numbers up:

1 + 2 + 2 + 5 + 8 + 9 + 99 = 126
Divide total by how many numbers: 126 / 7 = 18
Mean: (x-bar) x = 18
Median
• Median: “middle” value of an ordered dataset
– Arrange observations in order min to max
– If odd number of data, locate the middle observation
– If even number of data, average the two middle #’s
• What is the median of the following numbers:
1, 2, 2, 5, 8, 9, 99
Numbers are already ordered from smallest to largest
5 is the middle value
• What is the median of the following numbers:
1, 2, 2, 8, 9, 99
Numbers are already ordered from smallest to largest
2 and 8 are middle values, their average is 5
Mode
• Mode: most frequent value of a dataset
– Arrange observations in order min to max
– Count how many times a value occurs
• What is the mode of the following numbers:
1, 2, 2, 5, 8, 9, 99
Numbers are already ordered from smallest to largest
2 occurs the most often
• What is the mode of the following numbers:
1, 2, 2, 5, 8, 8, 99
Numbers are already ordered from smallest to largest
2 and 8 are occur the most often  bimodal (two modes)
Midrange
• Midrange: average of largest and smallest
– Arrange observations in order min to max
– Add min and max, then divide by 2
• What is the midrange of the following numbers:
1, 2, 3, 5, 8, 9, 99
Numbers are already ordered from smallest to largest
1 + 99 = 100 then 100 / 2 = 50
Resistant Measures
• A very small or large value of a distribution is
an extreme value when it is greatly separated
from the rest of the numbers
• Extreme values will affect some measures of
the center
– Some are called outliers
– Formally definition of outliers will come later
• Measures that are not affected are called
resistant
– Median and mode are resistant
– Mean and midrange are not
Which One Do We Use?
The mean and the median are the most
common measures of center
If a distribution is perfectly symmetric,
the mean and the median are the same
The mean is not resistant to outliers
Use the mean on symmetric data and
the median on skewed data or data with outliers
The mode is a common measure of center for
categorical data
You must decide which number is the most
appropriate description of the center...
TI Calculator Help on “Center”
• Press STATS, choose EDIT, and Edit again
• Enter your numbers in L1
(hit enter after each number)
• Press STATS, choose CALC, and
1: 1-Var Stats and hit enter
and 2nd 1 (L1) and then enter again
• A bunch of useful numerical calculations in
determining center will be displayed: mean,
min, median, and max
Distributions Parameters
Median
Mean
Mode
Mean < Median < Mode
Skewed Left: (tail to the left)
Mean substantially smaller than median
(tail pulls mean toward it)
Distributions Parameters
Mode
Median
Mean
Mean ≈ Median ≈ Mode
Symmetric:
Mean roughly equal to median
Distributions Parameters
Median
Mode
Mean
Mean > Median > Mode
Skewed Right: (tail to the right)
Mean substantially greater than median
(tail pulls mean toward it)
Density Curves
• Density Curves come in many different shapes;
symmetric, skewed, uniform, etc
• The area of a region of a density curve represents
the % of observations that fall in that region
• The median of a density curve cuts the area in half
• The mean of a density curve is its “balance point”
Describing a Density Curve
To describe a density curve focus on:
• Shape
– Skewed (right or left – direction toward the tail)
– Symmetric (mound-shaped or uniform)
• Unusual Characteristics
– Bi-modal, outliers
• Center
– Mean (symmetric) or median (skewed)
• Spread
– Standard deviation, IQR, or range
Mean, Median, Mode
• In the following graphs which letter
represents the mean, the median and the
mode?
• Describe the distributions
Mean, Median, Mode
• (a) A: mode, B: median, C: mean
• Distribution is slightly skewed right
• (b) A: mean, median and mode (B and C – nothing)
• Distribution is symmetric (mound shaped)
• (c) A: mean, B: median, C: mode
• Distribution is very skewed left
Central Measures Comparisons
Measure of
Central Tendency
Computation
Interpretation
Mean
μ = (∑xi ) / N
x‾ = (∑xi) / n
Center of gravity
Median
Arrange data in
ascending order
and divide the data
set into half
Divides into
bottom 50% and
top 50%
Mode
Tally data to
determine most
frequent
observation
Most frequent
observation
When to use
Data are
quantitative and
frequency
distribution is
roughly symmetric
Data are
quantitative and
frequency
distribution is
skewed
Data are
categorical or the
most frequent
observation is the
desired measure of
central tendency
Measures of Positions
• Quartiles (available thru 1-VarStats)
–
–
–
–
Divide the data set into 4 equal parts
Q1: 25th percentile
Q2: 50th percentile (Median)
Q3: 75th percentile
• Percentiles
– Divide the data set into 100 equal parts
Statistics and Parameters
• Parameters are of Populations
– Population mean is μ
– Population standard deviation is σ
• Statistics are of Samples
– Sample mean is called x-bar or x
– Sample standard deviation is s
Summary and Homework
• Summary
– Frequency distribution describes the data’s shape
• Listed in a frequency table
• Visually depicted in dotplot, histogram or stem-leaf plot
– Four typical measures of central tendency
•
•
•
•
Mean (aka average)
Median (middle value in ordered list)
Mode (most frequent data value)
Midrange (average of max and min)
• Homework
– pg 820–822; problems 3, 5, 7, 8