Elementary Statistics is the science of collecting, organising, analysing, and interpreting data. It helps describe, understand, and make inferences about real-world phenomena.
Statistics is used in various aspects of everyday life. It helps businesses make data-driven decisions, governments formulate policies, and scientists analyse results. Basic statistical knowledge allows us to be smart consumers of information.
For example, businesses may use statistics to analyse customer data and make decisions about product development and marketing. Governments can conduct surveys and employ statistical methods to inform public policy. Scientists apply statistics to determine if research results are significant. As individuals, we encounter statistics in news reports, advertisements, and research studies. Knowing some fundamental statistical concepts allows us to evaluate the information critically.
The two main branches of statistics are descriptive statistics and inferential statistics.
Descriptive statistics summarise and present data while inferential statistics make predictions and inferences about populations using sample data.
Descriptive statistics describe and summarise features of collected data. It provides simple summaries of the measures and distributions of the sample data.
For example, descriptive statistics can calculate measures like the mean, median, and mode to indicate the central tendency or typical values of a dataset. They can determine the range, variance, or standard deviation to show how spread out the data points are. Descriptive statistics create charts and graphs to visually summarise data.
Measures of central tendency indicate the centre point or typical value of a data set. They provide a single value that is considered representative of the entire distribution. The three main measures of central tendency are mean, median, and mode.
The mean, commonly known as the average, is calculated by summing all the values in a distribution and dividing by the total number of values. It is the balance point of the data. With a symmetrical distribution, the mean will be at the centre. The mean is sensitive to extreme highs and lows in data.
To calculate the mean:
Mean = Sum of all values / Total number of values
For example, the mean of the dataset {2, 3, 6, 7, 10, 12} is:
(2 + 3 + 6 + 7 + 10 + 12) / 6 = 7
The median is the middle value that separates the higher half and lower half of the data set when ordered from lowest to highest. It is the 50th percentile value. With an odd number of data points, the median is simply the middle value. With an even number, it is the average of the two central values. The median is less affected by outliers compared to the mean.
To find the median:
For example, to find the median of {5, 2, 8, 7, 3}, first order the data: {2, 3, 5, 7, 8}
The middle value is 5, so the median is 5.
The mode is the value that occurs most frequently in the data set. It can be thought of as the most “popular” number in the distribution. A set may have one mode, more than one mode, or no mode if all values appear equally often. Modes are best used for qualitative and categorical data rather than numerical data.
To find the mode, look at the frequencies of the values and identify the value(s) with the highest frequency.
For example, in the dataset {6, 3, 9, 6, 2, 9, 6}, the mode is 6 because it appears most often.
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Measures of dispersion, also called measures of spread, indicate how spread out or scattered the data points are within a distribution. They reflect the degree of variability present. Common measures of dispersion are range, variance, and standard deviation.
The range is the difference between the highest and lowest values in a dataset. It measures the span of the values. A small range indicates values are clustered close together, while a larger range reflects greater spread.
To calculate range:
Range = Highest value – Lowest value
For example, in the dataset {1, 5, 18, 3, 14}, the highest value is 18 and the lowest is 1.
The range is:
18 – 1 = 17
Variance measures how far each value deviates from the mean on average. It reflects how dispersed the data is about the mean centre point. A higher variance indicates greater variability and spread while a lower variance reflects data points closer to the mean.
Variance is calculated by taking the average of the squared differences between each value and the mean:
To calculate variance:
The formula is:
Variance = Sum of squared differences from mean / (Total number of values – 1)
Standard deviation is calculated from the variance to express variability in the same units as the original data. It’s the square root of the variance. Standard deviation measures the typical distance between a data point and the mean centre.
A higher standard deviation indicates a greater spread. A lower standard deviation indicates points are closer to the mean centre.
To find the standard deviation:
The formula is:
Standard Deviation = Square root of Variance
Probability deals with analysing and making predictions about the likelihood of events. It helps assess uncertainty and randomness. Probability provides insights about the chance or probability of certain outcomes or events occurring.
Some fundamental concepts in probability theory are sample spaces, events, and probability rules.
The sample space in an experiment contains all the possible outcomes that may occur. It is the set of all possibilities. To determine probabilities, we look at where events stand about the sample space.
For example, when rolling a standard 6-sided dice, the sample space of possible values is {1, 2, 3, 4, 5, 6}. Other examples of sample spaces could be:
Events are outcomes or subsets of the sample space. They are sets of elements that compose the sample space. Events can contain a single outcome or a combination of outcomes.
For example, rolling an even number on a die {2, 4, 6} is an event. Getting Heads when flipping a coin is an event. Drawing a King from a deck of cards is also an event.
Probability rules guide the computation of probabilities of different events occurring.
Some key rules are:
These rules allow the calculation of probabilities for single, combined, dependent, and independent events.
A probability distribution shows all the possible values or outcomes of a variable and the probability associated with each one. The total area under the curve equals 1.
There are two main types of probability distribution – discrete and continuous.
A discrete distribution deals with events that have a countable number of outcomes, like coin flips, dice rolls, or survey responses. The variable can only take certain discrete values.
Some examples are:
A continuous distribution deals with variables that can take any value on a continuum. There are infinitely many possible values rather than discrete outcomes. Values are not counted but measured.
Some examples are:
Continuous distributions are described by density curves rather than probability masses.
While descriptive statistics summarise a sample, inferential statistics make inferences about a wider population based on the sample data. Inferential methods allow drawing conclusions that extend beyond the data at hand.
It enables the estimation of population parameters based on sample statistics. We can test hypotheses and make predictions about a population too large
to practically measure in its entirety.
Inferential statistics relies on selecting representative sample data from a population. Both probability and non-probability sampling provide ways to obtain sample data.
Some sampling techniques are:
Probability sampling aims for representative samples while non-probability sampling does not guarantee randomness or equal inclusion probability.
Estimation methods use statistics calculated from a sample, such as the mean, to estimate corresponding population parameters like the mean. Confidence intervals provide a range of plausible values.
A confidence interval gives a range of values likely to contain the unknown population parameter at a stated confidence level. Wider intervals reflect less precision while narrower intervals indicate more accuracy.
The most common is a 95% confidence level. The 95% confidence interval suggests we can be 95% confident the true parameter lies within the range.
Hypothesis testing allows concluding a population by evaluating two competing hypotheses stated about the population.
The null hypothesis, denoted H0, states there is no effect or no statistical significance. It assumes no difference or relationship. The alternative hypothesis H1 states there is an effect or statistical significance present.
H0 is presumed true until evidence indicates rejection in favour of H1.
For example:
H0: There is no difference between the two population means
H1: There is a difference between the two means
A Type I error means rejecting the null hypothesis when it is true. A Type II error occurs when failing to reject a false null hypothesis.
The acceptable risk levels for each error type set the criteria for rejecting or not rejecting H0. A lower Type I error rate often increases the Type II error rate.
Correlation and regression techniques explore relationships between variables. Correlation examines if and how strongly variables are related. Regression models the relationships mathematically.
Correlation analysis measures the association between two variables to determine if they are related or independent. It does not imply causation. The correlation coefficient (r) indicates the strength of the correlation and whether it is positive or negative.
The Pearson product-moment coefficient (r) assesses linear relationships between continuous variables. It ranges from -1 to +1, with:
Spearman’s rho measures monotonic relationships between continuous or ordinal variables. It assesses if one value increases as the other increases, without requiring a linear relationship.
Regression analysis models the relationship between a dependent and one or more independent variables. It estimates how the dependent variable changes with the others.
Simple linear regression predicts a dependent value from one independent variable. It fits a straight-line equation to model the relationship.
Multiple regression predicts a dependent value from multiple independent variables. It finds the linear combination of variables that yield the line of best fit.
Regression quantifies the connections between variables and makes predictions. However, it does not determine causation from correlation.
Presenting data visually through charts, graphs, and other plots makes it easier to interpret key information instead of inspecting raw numbers. Effective data visualisation highlights patterns, outliers, trends, and relationships in data to provide insights.
Some common ways to visualise quantitative and categorical data are:
Histograms display numerical data distributions using bars. The bar heights represent the frequencies or counts of values. Histograms show patterns, centres, spreads, gaps, and outliers in data.
Bar charts use horizontal or vertical bars to visually compare different categories of data. The bar lengths show the quantities in each category. They help compare amounts across groups.
Pie charts show the categorical data as proportional slices of a circle. The arc lengths represent the percentages of each category relative to the whole dataset. Pie charts depict parts of a whole.
Scatter plots visualise relationships between two numerical variables by plotting each data point along the x and y axes. Patterns in the scatter of points reveal correlations and trends between the variables.
Effective visuals tailor the type of plot to the data being presented. Plots should accurately represent the data and highlight key features.
Delving Deeper into Practical Uses:
Statistics isn’t just about numbers on a page; it’s a powerful tool shaping the world around us. Let’s see how:
Quick Reference Glossary:
Need a refresher on key terms? This mini-glossary has you covered:
This introduction covered essential concepts in elementary statistics. The key topics included:
Learning the fundamentals of statistics allows us to summarize data, make estimations and informed decisions, identify relationships, and draw meaningful conclusions. It provides the basic skillset to analyze real-world information and unlock insights from data.
Many excellent resources for building statistical skills are available through textbooks, online courses, instructional websites, and real-world practice. Start applying your knowledge to datasets from everyday life. Use statistics to satisfy curiosity about the world.
If this article was to your liking feel free to browse some of our blog posts on LearningMole! Some great examples are Education Statistics or covering some fun facts about AI.
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