Descriptive Statistics: Unit 5

• Descriptive statistics is covered in Unit 5.

Introduction

• Records values from a dental exam of 30 people from an epidemiological study.
• It is difficult to ascertain the condition of their oral health with 240 data points for age, sex, cavities, fluorosis, and gingivitis.
• It is easy to know the general situation of some oral health parameters of the 30 people if you look at a second data set relating to these values.
• Statistics transforms the multiple data points into a small number of figures.
• Statistics is a math science with processes for recording, organizing, synthesizing, and analyzing data from observation.
• It is a tool used in many disciplines, including epidemiology.
• The first is descriptive statistics, which aims to:
• Arrange data into tables and graphs from any study
• Describe data with numerical indices and statistics.
• The second step is analytic or inferential statistics, which does the following:
• Analyze data for information beyond the study sample.
• Apply information to a population and make predictions.

Sampling

• It is impossible to study all the people in a demographic study.
• You can perform a sample of the subject of focus to perform research.
• If the chosen population represents the origin population, the results can be applied or inferred to the group of origin.
• Common terms in epidemiological studies:
• Population/universe/collective: a group with common characteristics chosen by the researcher
• Sample: a set of individuals chosen from the population to be studied
• The sample must represent the population by sharing the characteristics that define it.
• Researchers infer the results of the sample study to the total community population
• Sample size: Number of individuals in the sample.
• Statistical protocols determine the size based on population size and frequency of the phenomenon under study.
• Sampling: Selecting a sample of study from a population.
• Sampling unit: Group of individuals in the population from which the sample participants are chosen; can be the full population or specific groups.

Types of Sampling

• Sampling is a first critical step in any study.
• Conclusions cannot be applied if study participants are not selected well.
• A sample must reflect its population by presenting the characteristics that define the population.
• Study external validity will be affected if samples are not representative.
• Probabilistic Sampling:
• All individuals have the same chance of being chosen for the sample.
• Ensures the sample represents the population since selection biases are removed.
• Selection can happen with varied procedures.
• Simple Random Sampling:
• Done when the population size is not high.
• A complete list of all population members is available beforehand
• Assign a number to each individual and choose numbers randomly.
• Systematic Random Sampling:
• Requires a list of the study population.
• Calculate a sampling constant K by dividing the total population by the sample size (K = N/n).
• Choose a random number r between 1 and K.
• Take the listing of population individuals and count K individuals from position r, selecting the components of the sample.
• The initial list should be made without pre-defined criteria; the results could be erroneous if there is criteria pre-determination.
• Stratified Random Sampling:
• Most suitable when specific variables need respect.
• Researchers should select the sample considering the distribution of these variables in the source population.
• The distribution of variables must be known in the population when performing this sample type.
• Once you know how many individuals are needed from each stratum, choose randomly via simple sampling.
• Random Cluster Sampling:
• Sampling is from pre-selected groups, instead of the entire, complete population.
• Each of these groups is called a cluster.
• Is used for convenience to simplify the election.
• Commonly used clusters include hospitals, schools, or universities.
• Select a group of clusters that meets the population characteristics, then choose the components of the sample.
• Useful when the population is very large and spread out
• Only information on people in the cluster is required.
• Non-probabilistic Sampling:
• Individuals do not have the same probability of sample selection, making population inferences more difficult.
• Selected samples can be unrepresentative.
• This method is for studies when there are not enough resources for probabilistic sampling.
• Intentional Sampling: Choosing individuals for a sample based on the ease of access to a specific group.
• Snowball Sampling: Start with an initial number of individuals; then, ask them to find other people.
• Discretionary Sampling: The researcher directly selects the study sample from a group they believe will be useful based on their criteria.
• A representative sample from the origin population is needed.
• Probabilistic sampling ensures representativeness for external validity and result inferring.

Types of Variables

• Epidemiological research involves collecting information from sample individuals.
• "Data" collected from each person corresponds to the variables studied.
• Researchers should record the number of cavities in people, as well as the age, sex, and nationality of the test subject.
• Variables include:
• Age.
• Cavities present.
• Number of oral pieces.
• Sex.
• Nationality.
• The variable type will determine which statistical analysis/graphical representation can be used.
• Qualitative variables: Expressed through words with qualities.
• Each quality of the variable is a category.
• Variable sex can be male or female (two categories).
• Marital status can be: married, separated, single, or widowed (four categories).
• Variables can be nominal or ordinal:
• The sex variable is nominal, and the categories are expressed with names that don't have a type of order; it does not matter if men/women are listed first.
• Ordinal qualitative variables like "level of required studies" can be ordered logically as: infant, primary, and secondary.
• Quantitative variables: Expressed with numbers
• Can include the following:
• Continuous quantitative: The variables can take any value of the numerical scale.
• Includes age.
• Discrete quantitative: Can only take a limited number of values.
• Number of children (whole numbers only).

Ordering and Tabulating Data

• Value is determined for each sample subject once study variables are set.
• Data is recorded on a data collections sheet that is made for each study.
• A descriptive study on caries would include the following data:
  • Patient number.
  • Presence of caries (yes/no).
  • Number of decayed teeth.
  • Number of absent teeth.
  • Number of filled teeth.
  • Total number of teeth.
• After data is collected for all variables, researchers construct a database or matrix to register all data from the subject sheets.
• All database columns consist of a variable, and contain collected information.
• Next you need to build the distribution table of frequencies.
• The following should be included:
  • Total frequency (N): the total number of individuals analyzed.
  • Absolute frequency (f): Number of individuals with a value of the variable X1
  • Relative frequency (f): Quotient between each f1 and total population
  • Cumulative frequency of X1 (fa): Sum of the past frequencies for a value X.
  • The fa of the last X should match n.
• Each value X1 that the variable can take.
• The frequencies of each X1 are the second column (fi).
• The accumulated frequencies column is useful for median calculation.
• These frequencies are calculated by adding the frequency of an X with the frequencies of the prior X.
• Frequency of X1 =0 is 20 because there's no value before 0.
• In cases where data is vast, the table value is built organizing data in intervals (making analysis easier).
• The following protocol can be followed to construct intervals:
• Determine the range of X1 values (total number of different values): Range = (Ximax. – Ximin.) + 1.
• Determine the desired number of intervals
• Determine the amplitude or size of the intervals by dividing the range by the interval number.
• Build a new table by adding width to the amplitude that was calculated.
• the frequency of the upper interval limit, in this case the frequency of the value 5, which is 12, will always be included in the upper interval).
• A new Xi column calculating the marks of the new intervals will show on the new table.
• When working with frequency tables, each interval will be substituted by its value mark.

Graphical Representations

• After the compilation of collected and ordered data, research will begin recording each graphical value from each variable.
• Benefits to these representations are easy viewing and ideas on how the values are distributed.
• The display selected is based on variable type.
• Qualitative Variables:
• Rectangle diagram
• Sector diagram
• Pictograms
• Quantitative variables:
• Bar diagram
• Histogram
• Frequency Polygon
• Qualitative variables can be represented by rectangle diagrams containing the following information:
• Absolute frequency using the Y ordainment axis.
• X abcisas axis.
• Category number using number of rectangles depending on variable count.
• Equal rectangle width separated by same distance.
• Rectangle height corresponding to category display/frequency.
• Equal distribution in sectors that corresponds to the frequency.
• Quantitative discrete variables can be represented using a bar diagram.
• Similar to rectangles.
• Abscesses have a grade with the variable. Quantitative, continuous variables can be represented by histograms:
• The Y axis expresses the frequencies and the abscissa is the variable.
• Rectangles for each value of the consistent variable are next to each other

Measures of Centralization

• Include mode, arithmetic mean, and median.
• Designed to find the average values around which most data is centered.
• Only calculated for quantitative variables.
• Mode (Mo): most frequent value in the data
• Determined by searching for the value X with higher frequency in the frequency distribution.
• If there are two X values with same frequency, there are two modes.
• Arithmetic mean:.
• Sum of all X values in data, divided by the number of data.
• Median: X value in the central position, after ordering from lowest to highest value.
• This formula is used to find it easily with the following frequencies:
• Total values should be divided by 2
• Results for divisions can be found in frequency columns.
• Median values and values should correspond to the frequency, etc.

Measures of Dispersion

• How much the marks from the individuals' variable are close or far from the central tendency indexes.
• Insufficient for summarizing with variable type values.
• Central indexes will cause distribution types from 2 samples.
• Dispersion indexes have different interpretations and variables.
• Total amplitude: numerically distance between highest and lowest mark values.
• Interval is used to count marks.
• Formula is used.
• Variance: Media from each mark variance in regards to the value average
• Presented habitually like.
• Formula is used to calculate this.
• Deviation: The square root of the variance, which would yield an index with originally marked variables.
• Formula is used to calculate this.
• Coefficient of variation: Allows the comparison of data spreads between variables or distributions with different forms.
• The coefficient of variation is calculated from the normal deviation combined with the media.

Practical Case Statistics

• Descriptive statistical analysis is carried out on an epidemiological study to describe oral health using centralization indexes and dispersion.
• A descriptive epidemiological study was performed to determine the oral health of first-year students at a University of Madrid.
• A randomized, systematic sample was used to select the 30 components.
• Researchers gathered distinct variables on detention caries and gum condition.
• Studied data sheets that indicate variable order.
• Database shows column-numbered forms.