EBP Week 10 Ratio, Proportion, Percentage, Rate, Central Tendency and Variability

Ratio, Proportion, Percentage, and Rate

• Ratio:
  • Expresses the relative frequency of one dataset compared to another.
  • Example: The ratio of nurses to doctors in the health sector.
• Proportion:
  • Expresses the relative frequency of one dataset as a fraction of the whole dataset.
  • Example: Doctors in South Australia as a proportion of all doctors employed in Australia.
• Percentage:
  • The proportion expressed in hundredths.
  • Calculated by multiplying the proportion by 100.
  • Example: The proportion of doctors in South Australia multiplied by 100.

Example 10.1: Ratio, Proportion, and Percentage in the Health Sector

• Given:
  • Number of nurses in South Australia: 105,064
  • Number of doctors in South Australia: 28,030
• Ratio Calculation:
  • $Ratio = \frac{Number\ of\ Nurses}{Number\ of\ Doctors}$
  • $Ratio = \frac{105,064}{28,030} = 3.75$
  • Interpretation: For every doctor in South Australia, there are 3.75 nurses.
• Given:
  • Number of doctors in South Australia: 28,030
  • Number of doctors in Australia: 97,150
• Proportion Calculation:
  • $Proportion = \frac{Number\ of\ Doctors\ in\ South\ Australia}{Total\ Number\ of\ Doctors\ in\ Australia}$
  • $Proportion = \frac{28,030}{97,150} = 0.288524$
• Percentage Calculation:
  • $Percentage = Proportion \times 100$
  • $Percentage = 0.288524 \times 100 = 28.85\%$
  • Interpretation: 28.85% of Australian doctors work in South Australia.

Rates

• Quantify the level at which a health disorder or disease is present in a given population, usually over one year.
• Two types of rates:
  • Incidence rate
  • Prevalence rate

Example 10.2: Rates of Melanoma

• Melanoma: A malignant form of skin cancer.
• Incidence Rate:
  • Number of new cases divided by the total population at risk.
  • $Incidence\ Rate = \frac{Number\ of\ New\ Cases}{Total\ Population\ at\ Risk}$
  • In 2018:
    • New cases of melanoma: 11,000
    • Total population of Australia: 25 million
    • $Incidence\ Rate = \frac{11,000}{25,000,000} = 0.00044$
    • Per 100,000 Australians: $0.00044 \times 100,000 = 44$
    • Interpretation: In 2018, there were 44 new cases of skin melanoma for every 100,000 Australians.
• Prevalence Rate:
  • Number of existing cases divided by the total population at risk.
  • $Prevalence\ Rate = \frac{Number\ of\ Existing\ Cases}{Total\ Population\ at\ Risk}$
  • In 2018:
    • Existing cases of melanoma: 30,000
    • Total population: 25 million
    • $Prevalence\ Rate = \frac{30,000}{25,000,000} = 0.0012$
    • Per 100,000 Australians: $0.0012 \times 100,000 = 120$
    • Interpretation: In 2018, approximately 120 individuals were affected by skin melanoma for every 100,000 Australians.

Measures of Central Tendency and Variability

• Averages (mean) indicate where data are centered or clustered.
• Measures of Central Tendency:
  • Mean, median, and mode.
  • Focus on where the data are centered or clustered.
  • Represent the most typical or representative scores in a distribution.

Mean

• The sum of all values divided by the number of observations or sample size.

Median

• The single value in a dataset that divides all values into half.
  • Half the values are above, and half are below the median.
• Used when ordinal, interval, or ratio-scaled data are used.
• Cannot be used for nominal data (categories).
• Selected as the measure of central tendency when data are skewed.
• Less sensitive to outliers compared to the mean.
• Example: House auction sale prices; median sale price alleviates the impact of outlier sales.

Mode

• The most frequently occurring score in a distribution.
• Used for nominal data.
• Can be used for any level of scaling (nominal, ordinal, interval, or ratio).
• Generally not a satisfactory indication of central tendency because it only considers the most frequently occurring number.

Mean, Median, and Mode in Distribution

• Normal Distribution:
  • Mean = Median = Mode, all represented by the central line of the bell curve.
• Positively Skewed Distribution:
  • Mode Median Mean (from left to right on the curve).
• Negatively Skewed Distribution:
  • Mean Median Mode (from left to right on the curve).

Measures of Variability

• Variability indicates the dispersion or spread of values in a dataset.

Range

• The lowest and highest values in a dataset.
• $Range = Highest\ Value - Lowest\ Value$
• Problematic when there are extremely low or high values (outliers).

Variance

• Describes how far the numbers deviate from the mean (average variability about the mean).
• Calculated by summing the squared deviations about the mean and dividing by the number of cases.
• The higher the variance, the greater the overall deviation from the mean, indicating greater variability.
• Can overstate the true spread of scores.

Standard Deviation

• The square root of the variance.
• A more commonly used measure of variability for continuous data.
• The higher the standard deviation (relative to the mean), the greater the spread of data.

Interquartile and Semi-Interquartile Ranges

• Used when the median has been selected as the measure of central tendency (skewed data).
• Data is divided into quarters:
  • Quartile 1 (Q1): Lowest 25% of the numbers.
  • Quartile 2 (Q2): Next 25% of numbers (up to the median).
  • Quartile 3 (Q3): Second-highest 25% of numbers (above the median).
  • Quartile 4 (Q4): Highest 25% of numbers.

Relevance of Central Tendency and Variability in Health Practice and Research

• Central Tendency:
  • Helps understand what is normal, common, or expected.
• Variability:
  • Helps determine when an observation deviates from normal and by how much.
  • Indicates when an observation may become abnormal or pathological.

Body Mass Index (BMI)

• Used to estimate human body fat based on an individual’s weight and height.
• $BMI = \frac{mass}{height \times height}$
• Units:
  • Mass (kg)
  • Height (m)
• Example:
  • Mass = 70 kg
  • Height = 1.77 m
  • $BMI = \frac{70}{(1.77 \times 1.77)} = \frac{70}{3.1329} = 22.3435$
• A BMI value of 22.3435 falls into the 'Normal' category.

Obesity and the BMI Distribution Curve

• Study by Penman and Johnson (2006) investigated changes in the BMI distribution curve from 1990 to 2003.
• 1990 dataset:
  • Mean: 25.444
  • Standard Deviation: 4.8826
  • Sample Size: 1498
• 2003 dataset:
  • Mean: 27.7308
  • Standard Deviation: 6.11952
  • Sample Size: 4212
• Both distribution curves resemble a normal bell-shaped curve but are slightly positively skewed.
• The 2003 dataset is more positively skewed than the 1990 dataset.
• From 1990 to 2003:
  • The population became fatter.
  • Mean BMI increased from 25.44 to 27.73.
  • More people had BMI values in the high tail of the curve (BMI > 40).
• Implications for public health policy to reduce adult obesity.

Sampling

• Bell curves, distributions, and other concepts are the basics for understanding statistics.
• Key components in sampling and statistical inference.