Transcript Welcome to PSY206F - University of Cape Town

Measures of Central Tendency
• Chapter 4 of Howell
• Aim of descriptive stats: describe the
sample
• A lot of numbers in a sample, want to
summarise in a single number
• A lot of different ways to summarise a group of
numbers
• Eg. What was the movie like?
1
2
Describing the “bigness” of a sample
• Look at these samples:
• A: 4 6 5 3 4
• B: 50 30 45 37 48
• Sample A has smaller values than B
• It’s “central tendency” is smaller
• Useful thing to summarise about a sample
• Various ways to do it
3
Measures of central tendency
• Various ways to talk about the “overall
value” of a sample:
• Which value is the most “popular”
• Which value best describes the “bigness” of the
data
• Which value represents the halfway mark in the
dataset?
• Each of these is useful in different ways
4
Popularity: the Mode
• The mode of a data set is the value with the
highest frequency.
• Best describes nominal variables
• (the only central tendency measure for nominal)
• Eg: Which cooldrink is the most popular?
• Work out: Make a frequency table (F column
only), and select the value with the highest
frequency.
• If 2 or more values have the highest F, all are
modes.
Example: calculating the mode
• Work out the mode of x for a sample:
• abbcaacbbaccabaaabbbaaaccc
• Step 1 - work out a freq table (F column
only)
•
•
•
•
Value
a
b
c
F
11
8
7
• Step 2 - find the highest frequency
• a has 11, so the mode of x is ‘a’.
5
Half-way mark: the Median
• The median value is the value which lies
halfway in the dataset.
• Useful for ordinal, interval & ratio
• Eg: What mark did at least half of the class get?
• Work out: make a frequency table (F and Cum.
F and % Cum. F columns), and select the value
which is includes 50% of the % cumulative
freq.
• Only one value will ever by the median
6
Example: calculating the median
• Work out the median for sample:
• 4123432343
• Step 1 - work out the %Cum Freq on the
table
• Value
F
CF %CF
•
•
•
•
1
2
3
4
1
2
4
3
1
3
7
10
10%
30%
70%
100%
• The value 3 (70%) includes 50% of the
scores, so it is the median
7
8
Bigness - the Mean
• The mean is an estimate of the size of
values in a variable - the average value
• Useful for interval and ratio scales only
• Expresses the bigness of the variable OR
• Tells us what number to expect for the next
observation
• “What IQ is the next person who walks in here
likely to have?”
• Only one value can be the mean
9
Formula for the mean
• The mean is calculated using a formula:
• x bar is the symbol for the mean
• “sum all the observations of x, and divide
by n”
Example: calculating the mean
• Consider x, with a sample of observations:
• 12 15 22 37 11 40 32 14
• Step 1: Sum all the x values
• 12+15+22+37+11+40+32+14 = 183
• Step 2: Divide the sum by n
• n=8
• 183/8 = 22.9
• The mean is 22.9
10
Using each of them
• They each say something different
• mode: which value is selected most?
• median: which value represents halfway
through the data?
• Mean: what is the average value of
observations?
• For discrete variables, only the mean and the
median should be used
11