N318b Winter 2002 Nursing Statistics Lecture 2: Measures of Central Tendency and Variability

N318b Winter 2002 Nursing Statistics

Lecture 2: Measures of Central

Tendency and Variability

Nur 318b 2002 Lecture 2: page 2

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Institute for Work & Health

Today’s Class(es) mean, median, mode range, standard deviation, variance

<< 10 min break >> Some examples Applying knowledge to assigned

readings (Arathuzik; Hayman et al.)

focuses on determining and interpreting measures of central tendency and dispersion

Followed by small groups from 12-2 PM


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A Quick Review from Last Week - 1

Measurement ScalesNominal dataOrdinal dataInterval dataRatio data

Variable TypesDependentIndependent


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mean median mode

A basic cornerstone of most research statistics is that numeric data points tend to group together, usually in identifiable (predictable) ways – i.e.

they tend to congregate around a common value

Measures of Central Tendency

You should know what these three things are and how they differ from each other


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Mean

Most appropriate for ratio or interval data (i.e. continuous numeric data) but not if strongly skewed

= (x1 + x2 + x3 + xn ) / N

Where x1 + x2 + x3 + xn are independent data points and N is the total number of data points

Note: x1 + x2 + x3 + xn also written as “X”


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Some Properties of the Mean All data points contribute to its value Sensitive to extreme values Sum of deviations always zero i.e. (x-)=0 Sum of squared deviations at a minimum -

i.e. (x-)2 lower for mean than other terms Mean is algebraic thus it can be manipulated

making it more useful statistically When sample large enough (e.g. >25) it does

a good job estimating true population mean


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MedianMost appropriate for ratio data (i.e. continuously scaled) even if skewed

median = mid-point of distribution(i.e. the 50th percentile)

Divides the data into two equally sized groups (i.e. same frequency or count in each)


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Some Properties of the Median Typically not calculated as it is simply the

mid-point (but data must be sorted/ordered) Median not sensitive to extreme values thus

useful if data skewed Not used with nominal data since it requires

data to have an order Does not have to actually exist as a data

point (e.g. mid-point between adjacent data points)


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ModeTypically more useful for grouped data (i.e. ordinal or re-scaled continuous data)

mode = most common value

Has descriptive value but it is not a widely used statistic


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Some Properties of the Mode

Not calculated (but observed) If all values unique then no mode May be more than one mode (e.g.

bimodal, trimodal, etc.) Only measure of central tendency

for strictly nominal data


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Mean, Median and Mode

When distribution of data points is very even (i.e. normally distributed), then the three converge centrally

Mean, median, mode all in same position in a perfect distribution


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Mean, Median and Mode“real” data points rarely (never!) perfectly normally distributed thus typically some differences do exist

Mean

Median

Mode

Sample “left” skewed as mean is less than median


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Mean, Median and Mode

Age groups

Group 1 = (11, 12, 13, 13, 14, 15)

Mean affected by extreme value

1 = 13Group 2 = (11, 12, 13, 13, 14, 25) 2 = 17

Median is 13 – divides data in half

Mode is 13 – most common value


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Standard deviation Variance Percentiles Range

Viewed another way, most research statistics that are numeric data points also tend to vary from each other, usually in identifiable (predictable) ways – i.e. they tend to be spread out

Measures of Dispersion

You should know what these four things are and how they differ from each other

The “Flip-side”:


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Dispersion (or spread)

Two samples with the same mean can have very different dispersion

Sample B: More dispersed

Sample A: Less spread, SD of A < SD of B

Sample A measured more precisely?

Mean


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Standard deviation

2 SD includes about 95% of sample

-2 SD +2 SD

1 SD either side of mean includes about 68% of sample

-1 SD +1 SDMean


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SD = (x-)2 / N-1

Standard deviationKey indicator of the average point deviation from the sample mean

If SD is low relative to the mean then measure is more precise (see coefficient of variation in textbook)

SD - most important dispersion measure


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Variance:squared deviations from mean; important for later methods

Other measures of deviation

Range:maximum value - minimum value; useful for describing sample

Percentiles:Value above which and below which a certain proportion of the sample falls


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10 minute break !


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Assignment #1

Mean 8.510638Standard Error 0.19821Median 9Mode 9Standard Deviation 1.921719Sample Variance 3.693004Kurtosis 14.59094Skewness -3.777008Range 9.7Minimum 0Maximum 9.7Sum 800Count 94

Assignment #1 MarksExample 1


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What happens if we remove the zeros – i.e. the most influential (outlying) observations?


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Assignment 1 excluding zeros

Mean 8.888889Standard Error 0.071412Median 9Mode 9Standard Deviation 0.677478Sample Variance 0.458976Kurtosis -0.140281Skewness -0.935882Range 2.3Minimum 7.4Maximum 9.7Sum 800Count 90

Assign #1 – Zeros droppedExample 2


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Part 2: Application to the

Assigned Readings


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Arathuzik (1994)

Quick summary of the paper: – a pilot study examining the effects of a combination of interventions on pain perception, pain control and mood in metastatic breast cancer patients – pre-test / post-test experimental design– 3 groups enrolled with 24 (convenience sample) subjects randomly allocated to the intervention groups


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Q1. What do you think of the sample size

Only 8 per group gives little chance to accurately address hypotheses

What happens if you change age categories of only 2 subjects in Table 1? What about education level?

Small samples are unstable !

A few questions …


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Table 1 – Descriptive data


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Q2. How are the pain scales expressed?

Visual analogue scales with 0 being no pain and 10 being extreme pain

How are they treated in the analysis?

Table 2 - Continuous data

A few questions …

no painextreme

pain0 10

- this may make it even harder to see an effect since they are not very precise


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Hayman et al. (1995)

Quick summary of the paper: – matched pair analysis of twins to examine nongenetic influences of obesity on lipid profile and blood pressure both cross-sectionally (Phase 1, N=73 pairs) and longitudinally (Phase 2 , N=56 pairs)


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Q1. Describe the sample population in terms of race, age and sex? Did it change much over time?

Age: at Phase 1:

Sex:at Phase 1:

A few questions …

at Phase 2:

Race –

at Phase 2:

all white, both Phases

M=8.5 yrs, SD = 1.8 yrs

M=12.5 yrs, SD = 1.8 yrs

43.8% male, 56.2% female

44.6% male, 55.4% female


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Q2. how long was follow-up period

p278 - “median interval between measurements was 40 months

What does this mean?

Roughly half the time periods were longer than 40 months and half were less than 40 months (i.e. it was the “dividing line”)

A few questions …


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Table 2 – Contrasting twins


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Table 3 – Descriptive data for the follow-up study


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Next Week - Lecture 3: Graphs, Normal Curve and

Central Limit TheoremFor next week’s class please review:1. Page 13 in syllabus2. Textbook Chapter 2, pages 46-573. Textbook Chapter 3, pages 65-704. Syllabus papers:

i) Kilpack (1991) ii) Paulson & Altmaier (1995)


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Workshop Rooms:

H018, H19 and H9

MS016, MS017, MS018, MS022MS023, MS027, MS028, and MS029

All rooms are now confirmed for rest of the year so please go to the same room with your group as last time


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Valid Cumulative

Value Frequency Percent Percent Percent

20.00 1 4.0 4.0 4.0

22.00 2 8.0 8.0 12.0

23.00 3 12.0 12.0 24.0

23.50 1 4.0 4.0 28.0

24.00 3 12.0 12.0 40.0

24.50 1 4.0 4.0 44.0

25.00 4 16.0 16.0 60.0

25.50 2 8.0 8.0 68.0

26.00 5 20.0 20.0 88.0

26.50 1 4.0 4.0 92.0

27.50 1 4.0 4.0 96.0

28.00 1 4.0 4.0 100.0

Total 25 100.0 100.0

“In Group”Session – Q#1:3rd column not necessary – i.e. no missing data !


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A Quick Review from Last Week - 2

Summarizing Hypotheses

Null or Research? Directional or Non-directional? Causal or Associative? Simple or Complex?

Documents

N318b Winter 2002 Nursing Statistics Lecture 2: Measures of Central Tendency and Variability