Transcript Analysis of Quantitative Data

Analysis of Quantitative
Data
CENTRAL TENDENCY
DISPERSION
TABLES
GRAPHS
Analysis of quantitative data
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Specification:
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To analyse quantitative data including measures of central tendency
(mean, median and mode), measures of dispersion (range and standard
deviation) and graphs (bar charts and frequency tables)
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Central tendency: the tendency for the values of a variable to cluster
round its mean, mode, or median.
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Measures of central tendency
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Descriptive statistics include measures of central tendency which are
mode, median and mean average
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Data analysed in such a way that it is clearly displayed and understood
(tables and graphs)
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Examples:
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Mode: The most common score in a set of scores
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1 5 7 8 8 12 12 12 15 (mode is 12)
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Median: The middle score in a set of scores (in order)
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1 5 7 8 8 12 12 12 15 – median is 8
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1 5 7 8 8 11 13 14 15 20 – median is 9.5 (8+11)/ 2 =9.5
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Mean : arithmetical average (adding all the scores in the set and dividing
by the numbers of scores in a set)
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1 5 7 8 8 11 13 14 15 20 – mean is 10.2
Mean
Strengths
Mean
Limitations
Median
Strength
Median
Limitation
Mode
Strengths
Mode
Limitations
Practice!
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For each of the following sets of data (a) calculate the mean, (b)
calculate the median, (c) calculate the mode
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2, 3, 5, 6, 6, 8, 9, 12 ,15, 21, 22, 80
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2, 2, 4, 5, 5, 5, 7, 7, 8, 8, 8, 10
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2, 3, 8, 10, 11, 13, 14, 14, 29
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Extension: Which measure of central tendency is more appropriate for
each set of data? And why?
Measures of dispersion (Range and
SD)
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Calculates the spread of score in a data set
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Range – The range is a measure of dispersion, found by finding the highest
score/number and taking away the lowest score giving the difference
between the two.
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5 7 8 8 12 12 12 15 – range is 10 (15-5)
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Influenced by extreme scores so it may not always be useful
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Doesn’t tell us if the scores are bunched around the mean score or more
equally distributed
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1 7 8 8 12 12 15 16 55 (55-1 = 49)
Range
Strength
Range
Limitations
Standard Deviation
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A normal distribution of data means
that most of the examples in a set of
data are close to the average, while
relatively few examples tend to one
extreme or the other. SD calculates
the distance of a score from its group
mean.
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The example here shows the SD of IQ.
Each shaded block is 1 SD away from
the mean.
Measures of dispersion
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Standard deviation
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Measure of how far scores vary (deviate) from the mean average.
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The higher the standard deviation (SD), the greater the spread of scores
around the mean value (the data is more varied)
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A low SD indicates that the scores tend to be closer to the mean of the set
(data is less varied)
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The lower the SD – the more accurate and representative the mean of the
data set is.
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Add up the differences squared for all the scores and then divide that
number by the number of scores minus 1. Then finally, find the square root
and you have the standard deviation
Task
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Using the example on page 50 (on the next slide) and steps 1-5
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Calculate the standard deviation for these scores:
Score (x)
Mean(x̄ )
Deviation (x-x̄ )
Squared deviation
(x-x̄ )2
6
9
4
8
3
Step 4: n-1= ? (n:
number of scores)
Extension:
Help others if
you finish you
calculation
Sum of deviations
squared:
Standard deviation: ______
Standard deviation
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The SD does not necessarily indicate a right or wrong
value. It is a descriptive statistic – it describes the
distribution in relation to the mean.
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Another way of looking at Standard Deviation is by
plotting the distribution as a histogram of responses. A
distribution with a low SD would display as a tall narrow
shape, while a large SD would be indicated by a wider
shape
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Explain why a researcher may choose to use SD over the range to
measure dispersion?
Standard deviation
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Strengths
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SD is the most sensitive measure of dispersion as it is derived by using every
score in the data set
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It is a more precise measure of the spread of data from the mean as
compared to the range
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Limitation
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SD can be affected by extreme values (outliers)
Frequency tables
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A frequency table records the number of times a score
is found, rather than the score itself being displayed
against each participant
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Useful as the distribution can be seen in table form.
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A histogram or frequency graph can be used to
display the data
Table to show frequency of self rated
obedience scores
Self reported
obedience scores
Frequency
1
2
2
5
3
4
4
2
5
3
Histogram/frequency graph
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This type of graph is used to present the distribution of the scores. Unlike a
bar chart, where the bars a separated by a space, the bars on a
histogram are joined to represent continuous data rather than categorical
data.
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The values are presented on the x-axis and the height of each bar
represents the frequency of the variable
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TASK
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Sketch a histogram based on the data in the previous slide
Summary tables
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Summary tables represent measures of central
tendency and dispersion clearly
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Copy out the table
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Describe the features/trends of the data:
Table showing the self-rated obedience
scores of males and females (How
obedient the participant thinks they are)
Males
Females
Mean
obedience
rating
4.1
7.4
Median
obedience
rating
4
7
Mode
obedience
rating
4
6,7,9
Range of
obedience
ratings
3
3
Standard
deviation
1
1.2
Bar graph
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Useful to illustrate summary data
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Bar chars are used to present data from a categorical variable such as the mean,
median or mode. Categorical value is placed on the x axis and the height of the
bars represent the value of the variable.
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TASK Sketch a bar graph
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A bar graph showing the mean self rated obedience scores (from the table on the
previous slide)
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All graphs need a main title and axis also need titles
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Include a space between each bar (for bar graphs only)