1. Calculating & Reporting Healthcare Statistics
Second Edition
Chapter 10
Descriptive Statistics in Healthcare

2. Descriptive Statistics
Used to explain data in ways that are manageable and easily understood
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3. Descriptive Statistics
Rank
Denotes a score’s position in a group relative to other scores organized in order of magnitude
The position of the observation is more important than the number associated with it
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4. Descriptive Statistics
Quartile
Data organized in order of magnitude divided into four equal parts – each part is a quartile
First quartile is the lowest 25% of the data
Second quartile is the is up to 50% of the data
Third quartile is up to 75% of the data
Fourth quartile includes the remaining 25% of the data
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5. Descriptive Statistics
Decile
Represents data divided into ten equal parts
First decile is the lowest 10% of the data
The ninth decile includes the first 90% of the scores
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6. Descriptive Statistics
Percentile
Separate the distribution into 100 equal parts
A person who scores at the 54th percentile has a score greater than or equal to 54% of all the scores in the distribution
Called a percentile rank
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7. Descriptive Statistics
Percentiles
Help people understand their score relative to all scores from a group
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8. Descriptive Statistics
Measures of Central Tendency
In summarizing data, it is often useful to have a single typical or average number that is representative of the entire collection of data or specific population
Three measures of central tendency are frequently used: mean, median, and mode
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9. Descriptive Statistics
Frequency Distribution
Shows the values that a variable can take and the number of observations associated with each value
A variable is a characteristic or property that may take on different values
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10. Descriptive Statistics
Mean
The arithmetic average
It is common to use the term “average” to designate mean
To obtain the mean, add all the values in a frequency distribution and then divide the total by the number of values in the distribution
For example, seven hospital inpatients have the following lengths of stay: 2, 3, 4, 3, 5, 1, and 3
The frequency distribution in order is 1, 2, 3, 3, 3, 4, and 5
To construct a frequency distribution, all the values that the LOS can take are listed and the number of times a discharged patient had that particular LOS is entered
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11. Descriptive Statistics
Mean
In order to determine the mean we sum all the values in the frequency distribution and divide by the frequency
The total is 21
We arrived at this by adding
To arrive at the mean, divide 21 by the number of values (or frequency distribution) which in our case is 7
The mean (or average) equals 3 days
Formula
Total sum of all the values / Number of the values involved = X OR
Σ scores/ N
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12. Descriptive Statistics
Median
The midpoint (center) of the distribution of values
It is the point above and below which 50 percent of the values lie
Describes the middle of the data, literally
The median value is obtained by arranging the numerical observations in ascending or descending order and then determining the value in the middle of the array
This may be the middle observation (if there is an odd number of values) or a point halfway between the two middle values (if there is an even number of values).
To arrive at the median in an even-numbered distribution, add the two middle values together and divide by 2
The advantage of using the median as a measure of central tendency is that it is unaffected by extreme values
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13. Descriptive Statistics
Median
May be used in reporting data instead of the mean
Provides a more revealing representation of the data when there are outliers in the distribution
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14. Descriptive Statistics
Mode
The value that occurs with highest frequency
The most typical
It is the simplest of the measures of central tendency because it does not require any calculations
In the case of a small number of values, each value likely may occur only once and there will be no mode
The mode is rarely used as a sole descriptive measure of central tendency because it may not be unique because there may be two or more modes
These are called bimodal or multimodal distributions
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15. Descriptive Statistics
The choice of a measure of central tendency depends on the number of values and the nature of their distribution
Sometimes the mean, median, and mode are identical
The mean is preferable because it includes information from all observations
If the series of values contains a few that are unusually high or low, the median may represent the series better than the mean
The mode is often used in samples where the most typical value is preferred
The mode does not have to be numerical
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16. Descriptive Statistics
Measures of Variation
Variability
We also want to consider the spread of the distribution, which tells us how widely the observations are spread out around the measure of central tendency
The mean gives a measure of central tendency of a list of numbers but tells nothing about the spread of the numbers in the list
Variability refers to the difference between each score and every other score
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17. Descriptive Statistics
Range
The simplest measure of spread
The range is the difference between the largest and smallest values in a frequency distribution
The easiest to compute
It is the simplest, order-based measure of spread, but it is far from optimal as a measure of variability for two reasons
First, as the sample size increases, the range also tends to increase
Second, it is obviously affected by extreme values which are very different from other values in the data
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18. Descriptive Statistics
Variance
A frequency distribution is the average of the standard deviations from the mean
The variance of a sample is symbolized by s2
The variance of a distribution is larger when the observations are widely spread
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19. Descriptive Statistics
Variance
To calculate the variance, first determine the mean
Then, the squared deviations of the mean are calculated by subtracting the mean of the frequency distribution from each value in the distribution. The difference between the two values is squared (X – Xbar )2
The squared differences are summed and divided by N – 1
S2 = variance
 = sum
X = value of a measure or observation
Xbar = mean
N = number of values or observations
The term N – 1 is used in the denominator instead of N to adjust for the fact that the mean of the sample is used as an estimate of the mean of the underlying population
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20. Descriptive Statistics
Standard Deviation
The standard deviation is kind of the "mean of the mean“
Standard deviation (SD) is the square root of the variance
Because SD is the square root of the variance, it can be more easily interpreted as a measure of variation
If the SD is small, there is less dispersion around the mean
If the SD is large, there is greater dispersion around the mean
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21. Descriptive Statistics
Standard Deviation
To understand this concept, it can help to learn about what mathematicians call normal distribution of data
A normal distribution of data means that most of the values in a set of data are close to the "average," while relatively few values tend to one extreme or the other
The standard deviation is a statistic that tells you how closely all the observations are clustered around the mean in a set of data
When the examples are pretty closely gathered and the bell-shaped curve is steep, the standard deviation is small
When the examples are spread apart and the bell curve is relatively flat, that tells you have a relatively large standard deviation
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22. Descriptive Statistics
Standard Deviation
Normal distribution means that if the variable on every person in the population were measured, the frequency distribution would display a normal pattern, with most of the measurements near the center of the frequency
It also would be possible to accurately and summarily describe the population, with respect to variable, by calculating the mean, variance, and SD of the values
In a normal distribution, one SD in both directions from the mean contains 68.3 percent of all values
Two SDs in both directions from the mean contain 95.5 percent of all values
Three SDs in both directions from the mean contain 99.7 percent of all observations
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23. Descriptive Statistics
Not all distributions are symmetrical or have the usual bell-shaped curve
Some curves are skewed
Their numbers do not fall in the middle, but rather on one end of the curve
Skewness is the horizontal stretching of a frequency distribution to one side or the other so that one tail is longer than the other
The direction of skewness is on the side of the long tail
That is, if the longer tail is on the right then the curve is skewed to the right
If the longer tail is on the left, then the curve is skewed to the left
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24. Descriptive Statistics
Correlation
Measures the extent of a linear relationship between two variables
Can be described as strong, moderate, weak, positive (direct) or negative (inverse)
A correlation of 0 means there is no relationship between the variables
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