Category Archives: Blog

A theoretical physicist can sit at a computer with a pen and paper may not seem like a likely candidate to understanding how the brain works, but, according to physicists who study statistics and algebra, they can figure out revolutionary theories about how the nervous system works. When I met Princeton theoretical physicist William Bialek in 2013 during my undergraduate years at Indiana University-Bloomington, I asked him about the “magic” of physics and how scientist can capture abstract ways of thinking and apply them to how neurons in the brain work. Bialek’s book “Spikes: Exploring the Neural Code,” one of my inspirations to step into neuroscience research, and his work alongside other researchers in physics and mathematics can answer key questions in neuroscience.

Pairwise Interactions

Often in neuroscience we are confronted with a small sample measurement of a few neurons from a large population. Although many have assumed, few have actually asked: What are we missing here? What does recording a few neurons really tell you about the entire network? Correlations of neurons dominated large networks of neurons. Using Ising models from statistical physics, the researchers of Schneidman et al. 2006 looked at large networks and their ability to correct for errors in representing sensory data. They argue that correlations are due to pairwise, but not 3-wise interactions between neurons, although some might argue that closer inspection reveals otherwise. Pairwise interactions are how neurons forms pairs among themselves to act together. Their pairwise maximum entropy approach can capture the activity of RGB neurons effectively.

Using an elegant preparation retina on a micro electrode array (MEA) viewing defined scenes/stimuli, the researchers showed that statistical physics models that assume pairwise correlations, but disregard any higher order phenomena, perform very well in modeling the data. This indicates a certain redundancy exists in the neural code. The results are also replicated with cultured cortical neurons on a MEA. They noted a dominance of pairwise interactions. This would imply that learning rules depending on pairwise correlations could, on their own, create nearly optimal internal models describing how the retina computes codewords. The brain could, then, assess new events for their degree of surprise with reasonable accuracy. The central nervous system alone could learn the maximum entropy model from the data provided by the retina alone, but the conditionally independent model is not biologically realistic in this sense. Although the pairwise correlations are small and weak and the multi-neuron deviations from independence are large, the maximum entropy model consistent with the pairwise correlations captures almost all of the structure in the distribution of responses from the full population of neurons. The weak pairwise correlations imply strongly correlated states. 

If you modeled the cells independent from one another, they would form the Poisson distribution. The actual distribution is almost exponential, so this doesn’t fit well. For example, the probability of K = 10 neurons spiking together is ~10⁵ x larger than expected in the independent model. For this model, the specific response patterns across the population of neurons show that the N-letter binaries (patterns of 0s and 1s) differ greatly from the experimental results. These discrepancies show the failure of independent coding. The difference between prediction and empirical observation is anti-correlated in clusters of spikes. 

Instead, a group of neurons comes to a decision through pairwise correlations. These rates are predicted with >10% accuracy. The rates scatter between predictions and observations is confined largely to rare events for which the measurement of rates is itself uncertain.

The Jensen–Shannon divergence measures similarity between two probability distributions. This metric can be used to measure mutual information of a random variable to an associated mixture distribution, as the researchers did. In previous work, the researchers had used the same principle to a joint distribution and the product of its two marginal distributions and measure how reliably you can decide if a given response comes from the joint distribution or the product distribution. 

The fractions of full network correlations in 10-cell groups the maximum entropy model of second order plotted as a function of the full network correlation, measured by the multi-information I_N. The ratio is larger when I_N itself is larger, so that the pairwise model is more effective in describing populations of cells with stronger correlations, and the ability of this model to capture ~90% of the multi-information holds independent of many details. 

The Maximum Entropy Method

The most general description of the population activity of n neurons, which uses all possible correlation functions among cells, can be written using the maximum entropy principle as shown in the equation above for a probability p̂, Lagrange multipliers h_i, and J_ij, Z as the normalization constant, and the other variables representing each individual event probability. This method also uses Laplace’s principle of insufficient reason, which states that two events are to be assigned equal probabilities if there is no reason to think otherwise, and Jayne’s principle of maximum entropy, the idea that distributions are determined so as to maximize the entropy (as a measure of uncertainty) in a way consistent with given measurements.

For N neurons, the maximum entropy distributions with Kth-order correlations (K=1, 2, …N) can account for the interactions. Entropy difference (multi-information)  I_N = S₁ – S_N measures the total amount of correlation in the network, independent of whether it arises from pairwise, triplet or more-complex correlations. They found this across organisms, network sizes, appropriate bin sizes, Each entropy value S_K decreases monotonically toward the true entropy S : S₁ ≥ S₂ ≥,… ≥ S_N. The contribution of the Kth-order correlation is I_(K) = S_K-1 – S_K and is always positive. More correlation always decreases entropy. 

In a physical system, the maximum entropy distribution is the Boltzmann distribution, and the behavior of the system depends on the temperature, T. For the network of neurons, there is no real temperature, but the statistical mechanics of the Ising model predicts that when all pairs of elements interact, increasing the number of elements while fixing the typical strength of interactions is equivalent to lowering the temperature, T, in a physical system of fixed size, N. This mapping predicts that correlations will be even more important in larger groups of neurons.

The active neurons are those that send an action potential down the axon in any given time window, and the inactive ones are those that do not. Because the neural activity at any one time is modelled by independent bits, Hopfield suggested that a dynamical Ising model would provide a first approximation to a neural network which is capable of learning.

The researchers looked for maximum entropy distribution consistent with experimental findings. Ising models with pairwise interactions are the least structured, or maximum-entropy, probability distributions that exactly reproduce measured pairwise correlations between spins. Schneidman and the researchers used such models to describe the correlated spiking activity of populations of neurons in the salamander retina subjected to naturalistic stimuli. They showed that for groups of N≈10 neurons (which can be fully sampled during a typical experiment) these models with O(N²) tunable parameters provide a good description of the full distribution over 2^N possible states. 

They found the maximum entropy model of second order captures over 95% of the multi-information in experiments on cultured networks of cortical neurons. There would be implications for learning rules could be enough to generate nearly optimal internal models for the distribution of “codewords” in the retinal vocabulary and let the brain accurately evaluate new events for their degree of surprise.

Accounting for Cell Bias

The researchers noted they needed to account for the pairwise interactions and cell bias values. Interactions have different signs, the researchers showed that frustration would prevent the system from freezing into a single state in about 40% of all triplets. With enough minimum energy patterns, the system has a representational capacity, and the network can identify the whole pattern uniquely just as Hopfield models of associative memory do. The system would have a holographic or error-correcting property, so that an observer who has access only to a fraction of the neurons would nonetheless be able to reconstruct the activity of the whole population.

The pairwise correlation model also uncovers subtle biases in decision making. It will tell you about how they influence each other, on average. Pairwise maximum entropy models reveal that the code relies on strongly correlated network states and shows distributed error-correcting structure.

To figure out if the pairwise correlations are an effective description of the system, you need to determine if the reduction in entropy from the correlations captures all or most of the multi-information I_N. The researchers conclude that, even if the pairwise correlations are small and the multi-neuron deviations from independence are large, the maximum entropy model consistent with the pairwise correlations captures almost all of the structure in the distribution of responses from the full population of neurons. This means the weak pairwise correlations imply strongly correlated states. 

Other Effects

Intrinsic bias dominates small groups of cells, but, in large groups, almost all of the ~N² pairs of cells are significantly interacting (top). This shifts the balance so that the typical values of the intrinsic bias are reduced while the effective field contributed by other cells increases (bottom). In the Ising model, if all pairs of cells interact significantly with one another, you can limit the typical size of interactions by showing how J_ij changes with increasing N. There were no signs of significant changes in J with growing N with the values the researchers tested. 

Extrapolation

For weak correlations, you can solve the Ising model in perturbation theory to show that the multi-information I_N is the sum of mutual information terms between all pairs of cells, and I_N ~ N(N – 1) (left). This is in agreement with the empirically estimated I_N up to N = 15, the largest value for which direct sampling of the data provides a good estimate. Monte Carlo simulations of the maximum entropy models suggest that this agreement extends up to the full population of N = 40 neurons in their experiment (G. Tkačik, E.S., R.S., M.J.B. and W.B., unpublished data). The potential for extrapolation to larger networks of neurons can be shown through the error-correction that comes about (right). The error-correction emerges when figuring out how N-cell activity can predict (N+1)-cell activity. Uncertainty decreases by the number of cells. In a 40-cell population, three cells with spiking probability have an near-perfect linear encoding of the number of spikes generated by other cells in the network. Through these methods of becoming more and more accurate and robust, they showed findings that are similar to how single pyramidal cell spiking correlates with more collective responses. 

Challenges to the Model

The case of two correlated neurons has proven to be particularly challenging, because the Fokker–Planck equations are analytically tractable only in the linear regime of correlation strengths (r ≈ 0) and only for a limited set of current correlation functions. Some analytical results for the spike cross-correlation function have been obtained using advanced approximation techniques for the probability density and expressed as an infinite sum of implicit functions (Moreno-Bote and Parga, 2004, 2006). Similarly, the correlation coefficient of two weakly correlated leaky-integrate-and-fire neurons has been obtained for identical neurons in the limit of large time bins. 

Correlations between neurons can occur at various timescales. It’s possible, by integrating the cross-correlation function (xcorr in Matlab, correlate in numpy) between two neurons, to read off the timescale of the correlation (Bair, Zohary and Newsome 2001). This can help to distinguish correlations due to monosynaptic or disynaptic connections, which are visible at short timescales, with correlations due to slow drift in oscillations, up-down states, attention, etc., which occur at much longer timescales. Correlations depend on physical distance on the cortical map as well as tuning distance between two neurons (Smith and Kohn, 2008).

Decoding techniques of Ising model can be applied to simulated neural ensemble responses from a mouse visual cortex model with an improvement in decoder performance for a model with heterogeneous as opposed to homogeneous neural tuning and response properties. Their results demonstrate the practicality of using the Ising model to read out, or decode, spatial patterns of activity comprised of many hundreds of neurons (Schaub et al. 2011).

Discussion

The research seems to reflect general trends of “the whole is greater than the sum of its parts” or even “less is more,” both concepts in science and philosophy that date back centuries. I even emailed Elad Schneidman a few days ago about this, and he responded, “I think that this idea must predate the ancient greeks ;-)”.

Their work used the application of the maximum entropy formalism of Schneidman et al. 2003, to ganglion cells. The same way a group of neurons behaves differently than the sum (or combination) of each independent neuron gives the research leverage and potential for these systems-like problems of neurocomputation and emergent phenomena. 

The work in deriving an Ising model (or using a maximum entropy method) from statistical mechanics shows the importance of a priori proof work in using equations and theories to deduce “what follows from what.” It’s a great example of using the principles and methods of abstraction that mathematicians and physicists use in solving problems in biology and neuroscience. In my own writing, I’ve described this sort of attention to abstract models and ideas as relevant to biology in a previous blogpost.

In this paper, the researchers very well theorized which shortcomings and limitations their model would have and addressed them appropriately by fitting their model to experimental work. As a result, their research testifies to the power of computational and theoretical research in both describing and explaining empirical phenomena. 

Recreating the Results

Related Research

In that same year, Tkačik and other researchers would use the same recordings and use Monte-Carlo-based methods to construct the appropriate Ising model for the complete 40-neuron dataset. They showed that pairwise interactions still account for the observed higher-order correlations and argue why the effects of three-body interactions should be suppressed. 

They examined the thermodynamic properties of Ising models of various sizes derived from the data to suggest a statistical ensemble from which the observed networks could have been drawn and, consequently, to create synthetic networks of 120 neurons. They found that with increasing size the networks operate closer to a critical point and start exhibiting collective behaviors reminiscent of spin glasses. They examined more closely the appearance of multiple single-spin-flip stable states.

The method of using a maximum entropy model is equivalent to the method of Roudi et al. 2009, where they described a method of normalizing the the Kullback–Leibler divergence D_KL(P, P˜) (for P˜ approximation to distribution P, with the distance from the independent maximum entropy fit. The quality of the pairwise model comes from normalizing this by the corresponding distance of the distribution P from an independent maximum entropy fit D_KL(P, P₁), where P₁ is the highest entropy distribution consistent with the mean firing rates of the cells (equivalently, the product of single-cell marginal firing probabilities): Δ = 1 – D_KL(P, P˜)/D_KL(P, P₁) where Δ = 1 means the pairwise model perfectly fits the additional information left out by the independent model, and Δ = 0 means the pairwise model doesn’t improve at all compared to the independent model. 

In 2014, Tkačik and the researchers from Schneidman et al. 2006 published “Searching for Collective Behavior in a Large Network of Sensory Neurons” with K-pairwise models, more specialized variations of the pairwise models to estimate entropy, classify activity patterns, show that the neural codeword ensembles are extremely inhomogeneous, and demonstrate that the state of individual neurons is highly predictable from the rest of the population, which would allow for error correction. 

Barreiro et al. 2014 found that, over a broad range of stimuli, output spiking patterns are surprisingly well-captured by the pairwise model. They studied an analytically tractable simplification of the retinal ganglion cell mode, and found that in the simplified model, bimodal input signals produce larger deviations from pairwise predictions than unimodal inputs. The characteristic light filtering properties of the upstream retinal ganglion cell circuitry would suppress bimodality in light stimuli, thus removing a powerful source of higher-order interactions. The researchers said this gave a novel explanation for the surprising empirical success of pairwise models.

Ostojic et al. 2009 studied how functional interactions would depend on biophysical parameters and network activity that variations in the background noise changed the amplitude of the cross-correlation function as strongly as variations of synaptic strength. They found that the postsynaptic neuron spiking regularity has a pronounced influence on cross-correlation function amplitude. This suggests an efficient and flexible mechanism for modulating functional interactions.

In 1995, Mainen & Sejnowski showed that single neurons have very reliable responses to current injections. Nevertheless, cortical neurons seem to have Poisson or supra-Poisson variability. It’s possible to find a bound on decodability using the Fisher information matrix (Sompolinsky & Seung 1993). Under the assumption of independent Poisson variability, it is possible to derive a simple scheme for ML decoding that can be implemented in neuronal populations (Jazayeri & Movshon 2006).

The accumulation of noise sources and various other mechanisms cause cortical neuronal populations to be correlated. This poses challenges for decoding. You can get a little more juice out of decoding algorithms by considering pairwise correlations (Pillow et al. 2008).

References

Bair, W. “Correlated firing in macaque visual area MT: time scales and relationship to behavior.” (2001). Journal of Neuroscience. 

Barreiro, et al. “When do microcircuits produce beyond-pairwise correlations?” (2014). Frontiers. 

Bialek, William and Rangnathan, Ramek. “Rediscovering the power of pairwise interactions.” (2018). Arxiv. 

Hopfield, J.J. “Neural networks and physical systems with emergent collective computational abilities.” (1982). Proc. Natl Acad. Sci. USA 79, 2554–-2558. 

Jazayeri, M, Movshon, A. “Optimal representation of sensory information by neural populations.” (2006). Nature Neuroscience. 

Mainen, ZF, Sejnowski, TJ. “Reliability of spike timing in neocortical neurons.” (1955). Science. 

Moreno-Bote, R., and Parga, N. “Role of synaptic filtering on the firing response of simple model neurons.” (2004). Phys. Rev. Lett. 92, 028102.

Moreno-Bote, R., and Parga, N. “Auto- and crosscorrelograms for the spike response of leaky integrate-and-fire neurons with slow synapses.” (2006). Phys. Rev. Lett. 96, 028101.

Ostojic, et al. “How Connectivity, Background Activity, and Synaptic Properties Shape the Cross-Correlation between Spike Trains.” (2009). The Journal of Neuroscience. 

Pillow, Jonathan, et al. “Spatio-temporal correlations and visual signalling in a complete neuronal population.” (2008). Nature. 

Roudi, et al. “Pairwise Maximum Entropy Models for Studying Large Biological Systems: When They Can Work and When They Can’t.” (2009). PLoS Computational Biology. 

Schaub, Michael and Schultz, Simon. “The Ising decoder: reading out the activity of large neural ensembles.” (2011). Journal of Computational Neuroscience. 

Schneidman et al. “Network Information and Connected Correlations.” (2003). Physical Review Letters. 

Schneidman et al. “Weak pairwise correlations imply strongly correlated network states in a neural population.” (2006). Nature. 

Seung, HS, Sompolinsky, H. “Simple models for reading neuronal population codes.” (1993). PNAS. 

Shlens, Jonathan, et al. “The structure of multi-neuron firing patterns in primate retina” (2006). The Journal of Neuroscience 26.32: 8254-8266.

Smith, Matthew, and Kohn, Adam. “Spatial and Temporal Scales of Neuronal Correlation in Primary Visual Cortex.” (2008). Journal of Neuroscience. 

Tkačik, Gašper et al.  “Ising models for networks of real neurons.” (2006). arXiv.org:q-bio.NC/0611072. 

Tkačik, Gašper et al. “Searching for Collective Behavior in a Large Network of Sensory Neurons.” (2014). PLoS Comput Biol. 

There are three kinds of lies: lies, damned lies, and statistics.

Mark Twain, “My Autobiography”

Journalists need a good understanding of numbers. Tapping into the power of data would let them create more meaningful and effective stories. But making sense of numbers can be difficult. Reporting on data is often not as straightforward or manageable as other types of journalism. Writers need to separate signal from noise.

What’s more, researchers and writers need to know the context of data to draw appropriate conclusions. You can know everything about how candidates in an election fare against one another through polls and surveys, but, until you know the causes behind why people would vote that way, you can’t say much about those statistics. I’ve written more here on the nature of causation in the context of scientific research. This guide provides a logical, reader-friendly approach to writers wanting to harness the power of statistics.

Table of contents:

1. Know the numbers
2. Study the source
3. Remember the reader
4. Present the product

1. Know the numbers

Too often, writers throw around numbers not knowing what they mean. Here is a run-down of statistics terms you should know as a journalist:

Bayesian statistics

• If it rains, how does that affect which football team will win? This branch of statistics lets you figure out how likely something may occur based on how it depends on other factors. This lets you account for factors like false positives (when an experiment detects something that doesn’t exist) such as medical screening flagging false cues as cancer. With Bayesian models, you can account for different sources of information in putting together these conditional probabilities.
• Using Bayesian statistics to predict how likely future events are is “Bayesian inference.”

Beta distribution

• Using a pre-defined distribution, you can determine how well a baseball player will do at the beginning of a season even when you haven’t collected much data so far. Using her batting average of .270, you can create beta distribution (shown above with α = .81 and β = .219. The average is .270 and the standard deviation is σ² = .115.
• If you don’t know the exact probability something occurs, you can figure out how which probability is most likely by selecting it from a beta distribution of probabilities. You can use α and β to calculate the mean μ and standard deviation σ with:

• You’ll also find binomial distributions which use the same probability for all trials instead of letting it change.

Chi-square test (χ² test)

Suppose you wanted to find the relationship between being HIV positive and sexual preference. You survey 30 males and find the following data (in a contingency table):

	Sexual preference
	Male	Female	Both	Total
HIV+	4	2	3	9
Not HIV+	3	16	2	21
	7	18	5

Then, you can multiply the raw numbers and divide by the total to calculate how likely it is HIV+ determines sexual preference. This gives you expected values, different from the observed ones as shown below:

	Sexual preference
	Males	Females	Both	Total
HIV+
Observed (O)	4	2	3	9
Expected (E)	(9*7)/30  = 2.1	(9*18)/30 = 5.4	(9*5)/30=1.5
(O-E)	1.9	-3.4	-1.5
(O-E)^2	3.61	11.56	2.25
Not HIV+
Observed (O)	3	16	2	21
Expected (E)	(21*7)/30 = 4.9	(21*18)/30 = 12.6	(21*5)/30 = 3.5
(O-E)	-1.9	3.4	-1.5
(O-E)^2	3.61	11.56	2.25
				30

• If you have an expectation or prediction of what your results should look like, the chi-square test compares them to what you actually observe to tell you how well your predictions match what happens. This example is borrowed from David Stockburger at Missouri State.
• Researchers calculate this by finding the difference between observed and expected values using the formula χ² = (observed − expected)²/expected.
• Sometimes you’ll see the difference between observed and expected values referred to as the “residual.”

Confounding variable

• If you want to test if texting leads to an increase in crashes, you would want to make sure that text messages, not weather or traffic, cause the crashes. These extra variables the study doesn’t account for are confounding variables.

Controlled experiment

• If you give a drug to students to observe how it affects sleep, you should compare this group (the treatment group) to a controlled group, a set of students under the same conditions, but without the drug. This makes sure you can determine that it was the drug causing differences in sleep and not some other variable.

Correlation

• This tells you how well two variables are related to one another. Two stocks that change in similar ways to one another over time may be correlated.

Fisher’s exact test

	Cured	Not Cured	Total
Drug A	42	58	100
Drug B	14	86	100
Total	56	144

• Similar to the chi-square test, this test compares whether an outcome occurs using a contingency table (shown above). There’s no formal calculation, but it can give you an idea of the probability an effect occurs.

Histogram

• This tells you how data is distributed with the normal distribution shown above. The height of each bar shows you how many data points in the bin along the x-axis or how likely it is to fall in that bin. For probabilities, the area of the bins should equal 100 percent.

Margin of error

• When you make a measurement, the margin of error (sometimes called “uncertainty”) tells you how much that measurement can change due to other factors. You’ll typically find this in a range of a confidence interval, such as “40 percent +/- 1 percent.”
• If you’re polling a sample of people, the margin of error can tell you how close the sample is as representative of the entire population.

• Writer Robert Niles defines this as “1 divided by the square root of the number of people in the sample.”
• You can further break down error into bias and systematic error:
  • The same way standing on a weighing scale while wearing clothes makes you heavier, a bias creates an error based on how you measure something.
  • If, instead, the weighing scale itself isn’t calibrated properly, there’s a systematic error. This affects all results due to the nature of your measuring equipment itself.

Mean

• This is the average of a set of data points, generally written using μ. When dealing with statistics, keep your language precise to communicate the most effective message possible. If the average life expectancy in the U.S. is 79 years, know the standard deviation and sample size. You may not need to report those factors, but they’ll help you put your averages in context.
• When journalists write about the “average citizen” or the “average voter,” in most cases, they’re not referring to the strict mathematical definition of an average (the sum of each data point divided by the number of data points). Rather, journalists tend to refer to the “average” as a common, representative individual in a population. Keep in mind the statistical average only represents this “average individual” based on how the standard deviation and sample size.

Median

• If you listed your data points from highest to lowest, the value in the middle is the median. Because this doesn’t depend on how far spread out or varied the data points are, the median is, more or less, the “middle.” It doesn’t matter that the richest person in America makes four times as much as the middle-class. What matters is whether it’s greater or lesser than those in the middle.

• In some cases, the median can give you a more accurate idea of the “average” person in a population when reporting. Make sure you understand where the median falls in the space between the highest and lowest data point. That can tell you more about how the numbers are distributed.
• Paleontologist Stephen Jay Gould quoted Twain’s “damned lies” quote to argue that using the eight-month median survival time for peritoneal mesothelioma was misleading. Many people, like Gould who lived for two more decades, would live for years and take an optimistic, positive view of statistics in general.

Mode

• The mode is the number occurring most often. This simple and clean measurement can tell you who’s the most popular candidate in an election. You won’t see this much, but it’s helpful for comparing raw numbers against one another like sales figures.

Multiplication rule

• If there’s a 1/2 chance you’ll draw a red card from a deck and a 1/13 chance the card is a King, then there chance to draw aa red King is 1/2*1/13= 1/26. This holds for independent events.
• Keep track of how one event may affect the other. If you draw red card from a deck (with a 1/2 probability), the chance the next card is red is now 25/51 because you have one less red card in the deck.

Normal (or standard or Gaussian) distribution

• Imagine taking a set of heights in a population and graphing the heights on the x-axis with how many times they occur on the y-axis. If the data is “normally” distributed, then most people should fall around an average height with fewer and fewer heights farther away from this average as shown in the graph above. In the normal distribution, you can define this distribution using the mean and standard deviation with  σ as the standard deviation and μ as the mean.

• The normal distribution centers on the average and, with a greater standard deviation, it becomes more spread out in both directions. You most likely won’t report the normal distribution explicitly in a news story.
• The standard deviation lets you compare the mean to the distribution. About 70 percent of people are one standard deviation from the mean (in either direction), 95 percent are two standard deviations away and almost everyone, three standard deviations away. The Z-score also tells you how far away a data point is from the mean.
• If you wanted to test if a new psychiatric drug changed the frequency of mood swings, you might measure the number of mood swings in a population with the drug and a population without. If you found that the means of the two distributions are separated by a certain number of standard deviations, you can convert that to a p-value. The smaller the p-value, the more likely it is that the drug itself, not some random variable. This gives you a probability that the drug works.

Null hypothesis (H₀)

• To figure out if smoking truly causes cancer, scientists look for ways to show that “smoking doesn’t cause cancer” is false. This is a null hypothesis (H₀), usually used to show that there is no effect or no relationship between what you want to show. In the words of scientists, they look for ways to “reject the null hypothesis.”
• When creating a standard distribution, the p-value tells you how likely it is to reject the null hypothesis.

Quartile

• Split the data into four equally sized groups. The lowest quarter is the lower quartile, the highest quarter is the upper quartile and everything in between is the middle quartile. The range of the middle quartile is the interquartile range.

Range

• This is the highest value minus the smallest. Note that the range is a single number, not a range of numbers.

Regression

• Regression tests what causes something to happen. If smoking really does cause an increase in cancer, then you should see it if you make a graph of cancer prevalence vs. smoking like the graph above, usually with a line of best fit (shown in red). Given enough linear regressions, you can separate a scientific observation explain the relationship between into the variables that cause it.
• Keep in mind correlation does not imply causation. If you find that video game sales rise around similar times when violent crimes occur, you still need to show that one caused the other before drawing conclusions between the two. Otherwise things may be a coincidence or just a matter of randomness.
• You’ll see an R value (how well one variable explains the other) or an R² value (how well the model fits the data). The ANOVA (Analysis of Variance) creates an R² and whether the result is “statistically significant.”

Standard deviation

• The standard deviation is how widely values are spread apart or how much the data varies. This, along with mean, defines a normal distribution.
• You can calculate the standard deviation of the population the formula above with x̄ as the average of data points x over n number of data points with Σ the sum of each value (x_i – x̄)². If you want the standard deviation of a specific sample, use n-1 instead of n in the denominator because you only know the mean of that sample, not the population.
• The standard deviation squared gives you the variance. Sometimes researchers use “deviation” and “variance” interchangeably so keep in mind the difference.

Stochastic models

• These are ways to predict future data like financial portfolios or weather forecasts that depend on randomness. Using distributions like the normal or beta distributions, you can simulate what future data will look like and form predictions.

Variable

• Variables are anything that differs from person to person or sample to sample.

• Categorical variables are ways of labeling people into groups (like biological sex or state of residence), continuous ones lie on a scale (like age or temperature), qualitative ones use adjectives (like colors) and random variables are what scientists measure as outcomes of experiments (like flipping a coin).

2. Study the source

The Society for Professional Journalists dictates you should remain accountable and transparent, seek the truth and report it and act independently. In the context of numbers, this means remaining open and honest in data analysis, scrutinizing findings and mathematical methods and doing so free from anything that my interfere with an investigation. After you know the definitions of statistics, you need to know where those numbers came from. This means not only knowing how data was collected, but appealing to statistics in a way that reflects the current principles of journalism.

Writer William Davies argued the authority of statistics and the researchers who study them is declining. In a post-statistical society, journalists need to remain objective and skeptical to statistics while still appreciating them for what they are. It won’t be a battle between elite facts and populist feelings, but, rather, public rhetoric and the forces against it.

Remember to keep numbers in context of their original source or how they were measured. If someone asks where you got your information from or how a number was calculated, you should have an appropriate answer. If you’re reporting a p-value for biomedical study, which variables were measured? How does the standard deviation affect the certainty of the results? Make sure that, for whatever claim or argument a scientist has put forward in a study, you can be responsible for however you report on it.

As you become more statistically literate, you’ll naturally reevaluate how you reason. Becoming aware of common fallacies and pitfalls journalists fall into can make you more prepared to present accurate scientific findings. Be careful when you read a study suggesting that, because people are losing jobs, the economy must be doing poorly or that, if a study found no evidence on the link between fossil fuels and climate change that you conclude there’s evidence of absence. You can begin to see through the arguments that the majority of people saying something is true makes it true and, instead, take a more empirical approach to forming an opinion.

Much more sinister are those who prey on individuals without a strong statistical or mathematical literacy. Showing that the cost of attending college is a smaller percent of the national debt now than it was in the 1960s doesn’t show that today’s college students pay less for their education. As you study the context and nuances of scientific findings, you’ll become better prepared to separate signal from noise in these situations.

If there’s a 20 percent chance of rain, does that mean it will rain 20 percent of the time? If a medical procedure has a false positive rate of 1 out of 10 trials, how does that change its effectiveness? It’s easy to appeal to the authority of statistics and science without investigating for yourself. Check what experiments were performed or the historical use of tests like the Fisher’s exact test.

This way, you’re acting as both a writer and a researcher. The key here is to avoid resorting to phrases like “studies show” or “survey says,” and, instead, ask yourself if you really know what the scientific studies purport. Many times scientists will refer to terms like “standard deviation” or “variance” interchangeably so make sure you know what’s being reported.

3. Remember the reader

Now that you have a deep understanding of what you’re reporting and what it means, you need to put it in a context that a general audience can understand.

If you ask a drunkard what number is larger, 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.

Edward Frenkel, “Love and Math: The Heart of Hidden Reality”

In the quote above, how does drunkard arrive at the correct answer? The statistics are presented differently. In the rephrased question, he has a more “tangible,” usable way of understanding how the proportions of vodka would be arise from the distribution among people.

How well do you understand what you write? Try answering this question to find out.

Imagine you conduct a breast cancer screening using mammography in a certain region. You know the following information about the women in this region: The probability that a woman has breast cancer is 1 percent (known as “prevalence”). If a woman has breast cancer, the probability that she tests positive is 90 percent (“sensitivity”). If a woman does not have breast cancer, the probability that she nevertheless tests positive is 9 percent (false-positive rate). A woman tests positive. She wants to know from you whether that means that she has breast cancer for sure, or what the chances are. What is the best answer?

A. The probability that she has breast cancer is about 81 percent.

B. Out of 10 women with a positive mammogram, about 9 have breast cancer.

C. Out of 10 women with a positive mammogram, about 1 has breast cancer.

D. The probability that she has breast cancer is about 1 percent.

When German psychologist Gerd Gigerezner posed the question to about 1000 gynecologists, about 21 percent chose the correct answer, C. While that is a little worse than random guessing, I must admit that, on my first attempt, I failed to answer this question correctly, as well. Through his research, Gigerezner has crafted a theory of understanding statistics that would help us in situations like this.

Similar to Frenkel’s example of the fractions of vodka, psychologists like Daniel Kahneman and Gerd Gigerezner have shown that asking statistics questions in different ways can influence the ways we understand them. For example, when the information preceding the question is framed differently (as shown below), 87 percent of gynecologists answered correctly.

Assume you conduct breast cancer screening using mammography in a certain region. You know the following information about the women in this region:

• Ten out of every 1,000 women have breast cancer
• Of these 10 women with breast cancer, 9 test positive
• Of the 990 women without cancer, about 89 nevertheless test positive

In both examples (of breast cancer screening and of bottles of vodka), when we change from “conditional probabilities” to “natural frequencies,” we suddenly understand statistics much better. Like Gigerezner, I believe we can teach the appropriate way to interpret statistics, and, with the effect it has on our health and society, we have a moral imperative to do so.

This isn’t a simple case of deliberately communicating false information or lying about the statistics we use. While there may be agendas and conflicts-of-interests between professionals (including scientists), we simply don’t understand how to interpret statistics. And, in the field of medicine, this can have disastrous results. We make poor decisions about how long a patient may live, how prevalence of cancer among smokers, and understanding the harms and benefits of screening for breast cancer.

4. Present the product

Many ways of visualizing, illustrating or explaining statistics exist no matter the medium. Looking across FiveThirtyEight, The Guardian‘s Data section or other data journalism publications, you can find effective ways of communicating complicated concepts either to the audience of your publication or to colleagues. Use figures and graphs to explain take-home messages and conclusions from your reporting. Make sure they’re easy to read and follow.

Python and R offer ways of visualizing statistical findings with R providing much more extensive libraries for statistics than Python. My work in creating interactive network graphs, word clouds and even periodic tables show some examples. To produce a confusion matrix like the one shown below, you can use this code.

It’s a good idea to value openness and transparency with your code and work in creating visualizations. This gives other researchers and writers ways to check and re-examine what you’ve done. The chart below shows how much the University of California Santa Cruz Science Communication class of 2020 used Slack during their fall quarter (with its code here). Interactive graphs give the reader a better sense of data and let you communicate more information as effective as possible.

Make sure to perform statistical tests to confirm results from research when you report. In the movie “Rosencrantz and Guildenstern Are Dead,” the two protagonists flip a coin heads 92 times in a row. The chances this may happen is about 1 in 5 octillion. In a more realistic setting, the Dallas Cowboys have won 6 out of 8 coin tosses in the history of Super Bowls. In R, you can use a binomial distribution to return the value 0.109375.

probability <- .5 # Set the odds of getting heads to .5. 
wins <- 6 # number of winning coin flips
totalFlips <- 8 # total coin flips
dbinom(wins, totalFlips, probability)
> 0.109375

In the code, the comments are written with a # in front of them explaining what each line does. These comments are notes programmers write to explain things without affecting the code.

With enough coin tosses you can make a graph of how these probabilities are based on the number of heads and flips. When you only have two outcomes (heads or tails), it’s a Bernoulli distribution.

Not all visuals are created equal. Statisticians William Cleveland and Robert McGill found that people can tell differences between length and angles much more easily than shapes and colors. This means, where appropriate, you should use charts and plots that rely on lines slopes when possible and avoid pie charts.

No matter the code or plot you make, taking an independent investigative approach to statistics can let you harness the power of data in your stories. Becoming more savvy with numbers and calculations can let you present more accurate, verified findings. Though you can’t just drop statistics without context or understanding of how they came about, newsrooms and other workplaces for publications can use a more empirical approach in their findings in presenting scientific research for what it is. Whether its journalists themselves or a hired analyst creating statistical models of disease prevalence, they should adhere to the established standards of journalism.

Journalism emphasizes quick, easy-to-understand conclusions and messages. While some projects can require more complicated work flows such as Bayesian models, bootstrapping or exploratory data analysis, sometimes all that matters is whether an experiment worked or didn’t. In many cases, you simply don’t have the time or capacity to explain what a p-value or regression test is. Still, becoming statistically literate and understanding the mathematics behind calculations involved in research can make you all the more prepared in presenting stories. Being able to tell the difference between causation and correlation can save you from drawing false conclusions and make your arguments more justified on the basis of statistics. It can give you the power to check the work of others and move journalism into a domains of peer-reviewed, egalitarian work. In writing this guide, I hope to do so as well.