Data Presentation Methods

Data Presentation Methods

• Data is grouped or collated into frequencies to determine the rate of occurrence.
• Tabular presentation uses frequency tables
• Useful for both quantitative and qualitative data
• Graphical or diagrammatic presentation can use:
• Histograms or frequency polygons for quantitative or numerical data
• Bar or pie charts for qualitative or categorical data
• Dot maps for geographical mapping
• Summary indices include measures such as mean, median, and mode.

Tabular Presentation Definitions

• Class refers to one of the groups into which data can be classified
• Class Frequency (CF) is the number of observations (NOB) in the data set that falls into a particular class

Measures of Summarizing Data

• Measures of location or central tendency summarize data
• Measures of dispersion, spread, or variation summarize data
• Measures of partition summarize data and can be measures of dispersion

Measures of Central Tendency

• Describes the location of the center of a distribution of numerical and ordinal measurements
• Indicates what is typical for a group
• Identifies data values around which other values are distributed
• Locates the midpoint of a distribution
• Common examples are mean, median, mode, and midrange

The Mean (x-bar)

• Arithmetic average of the observations
• It is the most widely used average measure
• It is considered the most reliable and trustworthy measure
• It locates the center of gravity of a distribution
• It reveals where values for a group are centered
• It is suitable for numeric variables measured on interval or ratio scales when numbers can be added
• It cannot be used for nominal or ordinal variables because they cannot be added

Calculating the Mean

• Add up all the individual observations (summation ∑) and divide by the number of observations.
• Formula: Xbar = (x1 + x2 + x3 + x4 …+ xn) / N
• Simplified formula: Xbar = ∑x / N
• ∑x = summation of observation
• N = number of observations

Calculating the Mean for Grouped Data

• Find the class-mid mark for each interval
• Multiply the class mid-mark in each interval by corresponding frequencies, using the average of the class limits.
• Add the results across all intervals.
• Divide the results by the number of observations or total frequency.

Properties of the Mean (X)

• Affected by extremes of values
• All the other observations lie about it
• Makes use of all information
• The sum of the deviations of the values from the mean is always zero; derived by subtracting the mean from each value to form deviations (x minus xbar).
• ∑ (x- Xbar) = 0
• (x1 – Xbar)+ (x2 – Xbar) + (x3 – Xbar)…= 0

Advantages of the Mean

• Easy to calculate
• Uses all information in the distribution, making it reliable and accurate
• Amenable to statistical procedures and testing

Disadvantages of the Mean

• It can be unduly influenced by abnormal values in the distribution
• It should not be used with badly skewed distribution

The Median

• The middle number in an array of the data when the number of observations is odd
• The arithmetic mean of the middle numbers in an array of data when the number of observations is even
• It is the value above or below which half (50%) of the observations fall
• Is the bisector of histogram/ polygon
• Used for interval, ratio, and ordinal scales, but not nominal scales

Calculating the Median for Grouped Data

• Sample size (n) = 80
• Median position (n/2) = 40th
• Median class = 75-79
• Lower boundary (bL) = 74.5 (for median class)
• Frequency in median class, f = 20
• Cumulative above median class (F) = 36
• Class-width ( c ) = 5
• Formula: Median = bL + ((n/2 – F) / fmed) * c

Class Boundaries & Intervals

• A class or group boundary lies midway between the data values
• For data in the class or group is labelled 7.1 – 7.3
• The class values 7.1 and 7.3 are the lower and upper limits of the class.
• The different gives the class width.
• Class boundaries are 0.05 below the lower class limit and 0.05 above the upper class limit (because the figures are in 1Decimal place)
• The class interval/ width is the difference between the upper and lower class boundaries

Advantages of the Median

• Can be used with distribution of any shape, especially when data are skewed
• Easy to calculate and understand
• Offers better representations when there are outliers
• Not affected by extreme values, "the middle remains the middle”.

Disadvantages of the Median

• It does not use all information in the distribution
• Only takes into account one or two observations
• Provides no information about other observations.

The Mode

• The value/observation that occurs most frequently when observations are arranged in an array
• Most fashionable
• There can be several modes: bimodal, multimodal

Formula for Grouped Mode

• Mode is the value that has the highest frequency in a data set.
• For grouped data, class mode (or, modal class) is the class with the highest frequency.
• Formula: Mode= Lb + (∆1 / (∆1 + ∆2)) * XC
• Lb is the lower boundary of class mode
• ∆1 is the difference between the frequency of class mode and the frequency of the class before the class mode.
• ∆2 is the difference between the frequency of class mode and the frequency of the class after the class mode.
• C is the class width

Advantages of the Mode

• Easy to compute
• Not affected by extreme values
• Main usefulness is for calling attention to distribution in which the values cluster at several places
• Only average available for nominal scale

Disadvantages of the Mode

• It is not the best for biological or medical statistics
• Can result in several observations with the same frequency (multimodal)
• Least valuable

Mid-Range

• Calculated as: (Minimum + Maximum) / 2
• Not affected by extreme values
• Does not consider all information in the distribution

Choice of Measure of Central Tendency

• Depends on the shape/nature of the distribution is whether skewed to the left or right or a normal distribution
• Depends on the scale of measurement, is whether ordinal or numerical
• For continuous variation with a unimodal and symmetric distribution, the mean, median, and mode will be identical and lie on the same plane
• When the distribution is skewed, the median may be a more informative descriptive measure to use than the mean, as it is not affected by extreme values
• For statistical analyses and tests of significance, the mean is better whenever possible since it includes information from all observations, and its theoretic properties provide for more powerful statistical tests

Notes on Mean vs. Median

• If mean = median, the distribution is symmetrical
• If mean > median, the distribution is skewed to the right (positive)
• If mean < median, the distribution is skewed to the left (negative)

Fundamentals of Biostatistics

• Fundamentals of Biostatistics 8th Edition Bernard Rosner. (Chapters 1 &2). http://galaxy.ustc.edu.cn:30803/zhangwen/Biostatistics/Fundamentals+of+Biostatistics+8th+edition.pdf