4' Summary Statistics

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4. Summary Statistics

• Reading RB 50-70 R 82-89
• In section 3 we represented the data in some
  graphical forms.
• This sometimes involved some summarisation
• In this section we look at some summary measures
  of the data, i.e. measures that says something
  important about the data but do not represent the
  full data
• The following are some example summary
  statistics
• Average the average age of patients undergoing
  heart operations is 57
• Range each operating theatre is used between 4
  and 11 times each day
• Maximum nurses in Antrim hospital earn a maximum
  of 32,000 a year
• The main summaries used are the average and the
  spread.

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4.1 Average (Measure of Location)

• The term "average" refers to any measure of
  location
• There are 2 commonly used measures of location
  the mean and the median
• Mean
• sum all the data values and divide by the total
  number of data items in the sample
• Median
• arrange the data in order of size and pick the
  middle value
• If the number of data items is odd, pick the
  middle data item value as the median
• if the number of data items is even, pick the
  mean of the two middle values as the median
• Mode
• The sample mode is defined as the value which
  occurs with the highest frequency. Modes may not
  be unique.

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4.1 Average (Measure of Location)

• Examples of Mean and Median
• 10 people were asked how many cups of tea they
  drank yesterday
• Person 1 2 3 4 5
  6 7 8 9 10
• No. cups 2 3 1 5 3
  3 8 0 2 1
• Mean (2315338021)/10 2.8
• Median
• Order the data 0 1 1 2 2 3 3 3 5 8
• Median the average of the 5th and 6th
  (23)/2 2.5
• Note It only makes sense to calculate the mean
  or median of numerical data, e.g. can you talk
  about the average country of origin of patients?

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4.1 Average (Measure of Location)

• When to choose the mean and when the median?
• Symmetric sample ? mean
• Skewed sample ? median
• Why should you not use the mean when the data is
  skewed? Consider calculating the mean salary in a
  private nursing home where the owner pays himself
  110,000 a year and the other 9 employees only
  get 10,000 each. Does it make sense to say that
  on average people in the nursing home earn this
  much? What is the median?
• Mean (110000100009)/10 20,000
• Median 10000
• If you have doubt about whether a sample is
  sufficiently symmetric, a safer option would be
  to use median.

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4.2 Spread

• Spread
• is a measure of how "spread out" the data is
  about the average.
• Samples with the same average may have very
  different spreads.
• Here are two measures of spread
• If we use the mean
• Symmetric sample ? Standard Deviation (SD)
• If we use the median
• Skewed sample ? Semi-Inter-Quartile Range (SIQR)
• SD is the partner of the mean, and SIQR is the
  partner of the median

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Standard Deviation (SD)

• To calculate SD
• calculate mean
• subtract mean from each data value (x-mean)
• square each difference (x-mean)2
• sum the squares of differences ? (x - mean )2
• divide by (sample size - 1) ? (x - mean )2
  (This is variance)
• take square root
• Example Cups of Tea

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Semi-Inter-Quartile Range (SIQR)

• To calculate SIQR
• place values in order of size
• determine the two inter-quartile values Q1 and Q3
  (Q2 is median)
• SIQR is (Q3-Q1) / 2
• Example Sups of Tea
• Data in order 0 1 1 2 2 3 3 3 5 8
• Q2 2.5, Q1 1 Q3 3 --gt SIQR (3-1)/2 1
• Cups of Tea example summary
• Median 2.5 SIQR 1
• Mean 2.8 SD 2.29

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4.2 Spread

• Comments
• Note that extreme values affect the SD much more
  than the SIQR. Also, extreme values affect the
  mean more than the median
• If the extreme value 8 is omitted, the new mean
  and median should be quite close. What are they?
• Mean 2.22 median 2
• SD sqrt (2.57) 1.60 SIQR 1
• if the extreme values 8 and 0 are omitted
• mean2.5 median2.5 SD sqrt(1.82) 1.35
  SIQR0.75
• Note It never makes sense to calculate an
  average or spread for nominal data.

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5. Cross-tabulation and Data Coding

• 5.1 Cross-tabulation
• Sometimes your sample may contain different
  groups, e.g. you measure salary grades of men and
  women. You can view this as simply a set of
  grades but you may instead be interested in
  examining the data values for the groups
  separately, e.g. to see if there is a difference
  between the grades of men and women.
• Of course to look at the data in this way you
  must have recorded for each grade whether it was
  a man or woman. Essentially then you have
  measured two variables for each individual in the
  sample
• sex (male or female) - a nominal variable
• grade (1, 2, 3 or 4) - an ordinal variable
• A cross-tabulation is simply a table listing the
  frequencies of each data value for each group,
  e.g. the number of people at each grade for each
  sex. (summarising data)

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5.1 Cross-tabulation

• For example the cross-tabulation might look like
• 1 2 3 4 total
• male 18 12 15 15 60
• female 26 3 1 0 30
• total 44 15 16 15 90
• Notice where the row and column totals come from.
  
• Note that there should usually only be few values
  that each variable can take.
• According to this table
• how many females are in either grade 2 or 3? 4
• what percentage of males are in grade 1?
  18/44
• what overall percentage of staff are in grade 4?
  15/90

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5.2 Expected values

• Note how the totals are calculated from the
  frequencies.
• Suppose we look at the totals by themselves for a
  moment, e.g.
• 1 2 3 4 total
• male 60
• female 30
• total 44 15 16 15 90
• Suppose that we were just given this information
  and we were asked how many males at grade 2 we
  would expect there to be.
• How can you calculate your expectations?

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5.2 Expected values

• We might think about this as follows
• Since there are twice as many men as women and
  there are a total of 15 at grade 2 we would
  expect that 10 of them are men and 5 are women.
• For any position in the table you get the
  expected value for a particular row and column
  position from
• row total X column total
• overall total
• Use the above formula to calculate all the
  expected cell values
• Check that you get the same number in the "males
  at grade 2" position that we calculated above.

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5.2 Expected values

• We can include our expected values with the
  observed values in the table (expected values in
  italics)
• 1 2 3 4 Total
• male 18 12 15 15 60
• 29.33 10.00 10.67 10.00
• female 26 3 1 0 30
• 14.67 5.00 5.33 5.00
• Total 44 15 16 15 90
• Observations and Expectations can be compared
• From this cross-tabulation with expected values
  we can see that more males than expected are at
  the higher grades and correspondingly less women
  than expected are at the higher grades.

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5.3 Coding values to form groups

• An example from the paper by Rukhholm et. Al
• A survey involved asking many patients 20
  questions about how anxious they were about... .
  The answers were coded as follows
• 1 Not at all, to, 4 Very much
• Combining the answers gave a scale from 20 to 80
  for anxiety. The sex and age of each patient was
  also recorded.
• The sample was first divided into different age
  groups by coding the ages as follows
• MTBgt CODE (1834)1 (3551)2 (5268)3 (6985)4 c1
  c6
• The means and SDs of each age group was then
  calculated in order to examine if the anxiety
  level was different for different age groups.
• Next each of these 4 age groups was divided into
  a male and female group. This gives 8 groups in
  all.
• Male18-34 Female18-34 Male36-51 ..
• This allowed the mean anxiety level of each age
  group for a given sex to be compared with another
  such group.

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5.3 Coding values to form groups

• Another example
• Suppose we have the following data
• Age BScN(1)/BScPDN(2)
• 20 1
• 23 1
• 32 2
• 28 2
• 24 1
• This gives us a BScN group with data 20, 23 and
  24 and a BScPDN group with data 32 and 28.
• Supose we divide the ages into categories less
  than 26 and more than 26. (Collapsing for
  summarising)
• Draw the cross-tabulation table (including
  expected frequencies) that you get from this data.

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5.3 Coding values to form groups

• An example (cont.)
• BScN BSCPDN Total
• Age lt 26 3 0 3
• Age ? 26 0 2 2
• Total 3 2 5
• Notice how we can take a ratio variable (such as
  age above) and by coding it into groups it can be
  treated as an ordinal variable. After being
  grouped, age gave rise to a new variable with
  values 1 and 2 (corresponding to less than 26 and
  greater than 26).