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You are probably most used to thinking about the weather in your day-to-day lives. Should I bring an umbrella to class today? Is it warm enough for me to wear shorts? Can I gamble on that snowstorm cancelling my exam? The term “climate” may imply something completely different than what you see on the 6 p.m. news every night, but they are intricately related, differing in terms of time scale and predictability.

Climate is the “synthesis of the weather in a particular region.” Put another way, “Climate is what you expect ... weather is what you get!

Essentially, we can consider climate as the average outcome of all the weather at a particular location. Imagine you are playing a game like Monopoly that requires two dice. Any single roll of the dice could be something different – a 2, a 6, a 9. But over many, many rolls, a pattern begins to emerge. 2 and 12 happen, but are very rare. 6s, 7s, and 8s keep coming up time and time again. Any single roll of the dice can be thought of as “weather” -- unpredictability factor in the dice roll -- while the the regularity of dice rolls establishes a predictable pattern that informs how frequently those numbers can come up, just as climate does with weather data over decades.

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Before we delve into some of the more scientific details about climate, it’s a good idea to understand “the big picture.” Climate impacts everything! This could be you and me, animals, plants — things that are living. But its influence doesn't stop there; it extends to non-living elements as well. Ever strolled along a beach adorned with smoothed stones? Those stones owe their formation to relentless weathering caused by tiny sand particles. And guess what? These sand particles get whipped up by waves, courtesy of atmospheric winds—proof that climate is constantly at work!

This fjord in Norway is the result of a glacier cutting through a valley. Glaciers are consequences of climate — they form over time when snowfall exceeds its removal, leading to a slow-flowing mass of ice.

Credit: Geirangerfjord, I, Fgmedia, CC BY 2.5

But a common question posed to climate scientists is "so what? How does climate impact our everyday lives?" You might be astonished at the multitude of sectors influenced by climate. One critical aspect of climate's significance lies in agriculture. Statistics of temperature and precipitation influence crop growth and the success of farming practices. Deviations from climate norms can lead to crop failures, affecting food security, livelihoods, and global food prices. Conversely, a stable and predictable climate is vital for sustaining agricultural systems that feed billions of people worldwide.

Water availability is another important motivation to understand climate. Climate influences the occurrence of droughts and floods, which have profound consequences for communities and ecosystems. Prolonged droughts lead to water scarcity and conflicts over resources. Conversely, intense rainfall and flooding can result in damage to infrastructure, displacement of populations, and the destruction of ecosystems. If we think about these events as “rolls of the dice,” climate plays a key role in determining the frequency and severity of these events, making it an important factor in water resource management and disaster preparedness.

Crops may be stressed during periods of low temperatures, leading to reduced supply and high prices.

Credit: David Condos, Kansas News Service, CC BY 4.0 DEED

Climate impacts ecosystems directly by altering temperature and precipitation patterns, which in turn affect species distribution, migration, and the health of ecosystems. Maintaining ecological balance is essential for biodiversity and human well-being. Therefore, climate stability is crucial for preserving these intricate relationships and ensuring the resilience of our natural world.

In the context of sustainability, climate considerations are central. Sustainable practices, whether in agriculture, energy usage, or urban planning, must account for climate impacts to ensure they do not harm the environment or compromise future generations' well-being.

Moreover, climate is intricately linked to energy usage and planning. The type and quantity of energy we use have direct consequences for greenhouse gas emissions, which are a primary driver of climate change. Changes in temperature lead to changes in demand for heating and cooling and changes the stress on our electric grid. Transitioning to renewable and more efficient energy sources is crucial to mitigate climate-related risks.

Climate also influences air quality, as temperature and weather patterns affect the dispersion of pollutants and the formation of smog. Poor air quality, exacerbated by climate change, can have detrimental effects on human health and the environment.

Lastly, sea level rise, driven by the warming of the planet and the melting of polar ice caps, poses a significant threat to coastal communities and infrastructure. Understanding and mitigating sea level rise is essential for the long-term resilience of coastal regions worldwide.

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You may have heard of the three-legged stool analogy. It is often used to describe a concept or situation where three essential components or factors are critical. If one of the legs is missing, you are no longer sitting on a stool, but rather toppled over on the floor!

Three-legged Stool Example

Scientists understand climate by using three tools at their disposal. These three legs are:

Observations

“Observations” refer to the empirical data collected about the Earth's climate system through various methods. We’ll talk about these soon, but they can include satellites, weather stations, ocean buoys, and ice cores, among many others. These measurements can be straightforward, like temperature readings from a thermometer in your backyard, or incredibly complex, requiring highly specialized instruments to gauge parameters such as aerosol concentrations in the atmosphere or salinity levels in the ocean. Observations provide the essential raw data that feed into analyses, studies, and models, giving us the foundational "truth" about the evolution of the climate system. They offer a snapshot of real-world conditions, allowing scientists to validate hypotheses, calibrate models, and assess the current state of the climate system.

Models

“Models” are mathematical representations of the climate system built using the fundamental laws of physics, including conservation of mass, conservation of energy, and more. They can range from very simple “back-of-the-envelope” calculations you can do on a bar napkin to vast multi-million-line programs run on massive supercomputing systems that cost millions, if not billions, of dollars. Models serve as a virtual laboratory for experimenting on Earth, allowing scientists to simulate and analyze various scenarios to understand potential climate outcomes. These simulations are vital for predicting future climate conditions, assessing potential impacts, and developing strategies for mitigation and adaptation.

Experiments

“Experiments” in climate science involve controlled, often highly specialized, tests designed to isolate and study specific components or interactions within the Earth's climate system. These can range from lab-based studies that quantify fluxes from atmosphere-ocean interactions to field experiments that measure emissions from agricultural sources or the effect of land-use changes on emissions. Some experiments utilize advanced technologies like radiative forcing chambers to understand the response of atmospheric gases to various forms of radiation. Other experiments might employ techniques like stable isotope analysis to trace the origins and fates of specific elements within climatic cycles. The aim is to establish causality, validate or refute theories, and gather data that can be incorporated into climate models for more accurate predictions. Experiments are critical for advancing our understanding of complex climate processes, enabling scientists to tease apart the myriad factors contributing to climate change, and ultimately helping us make informed decisions for a sustainable future.

All of the legs complement one another, and focusing too much on one leads us to lack a large-scale overview of the Earth’s climate.

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Since climate can be thought of as the statistical eventuality of individual weather snapshots, scientists tend to use quantitative metrics to describe aspects of the Earth system. Measures of central tendency are statistical indicators that represent the center or the average of a data set and are crucial for summarizing a dataset with a single value, facilitating easier understanding and comparison. The three main measures of central tendency are the mean, median, and mode.

This is a lot of text; maybe you could use a more visual explanation of “the three Ms.” Watch this quick YouTube video (11:04 minutes) to gain a general understanding of mean, median and mode, then read over the text below for more details.

Video: Math Antics - Mean, Median and Mode (11:03)

Mean

The most used measure of central tendency in climate (and the one you have almost certainly used the most in your day-to-day life) is the mean, sometimes referred to as the arithmetic average or, more simply, the average. It provides a singular numerical representation of the quantitative data set’s center. The process of computing a mean is relatively straightforward, beginning with the summation of all individual elements or data points within the set. For a given set of n elements, denoted as x1, x2, …​, one starts by adding all these elements together, generating a cumulative sum of all values present in the set.

After obtaining the cumulative sum of the data set, the next step in calculating the mean involves dividing this sum by the total number of elements or data points in the set, which is represented as 'n'. Mathematically, this can be expressed through the formula:

$\overline{x}=\frac{1}{n}(x_1+x_2+\cdots +x/n)$

For example, let’s say we have five temperature measurements, each in Fahrenheit: 69, 74, 67, 74, 66. The sum of the elements (210) would be divided by the number of elements in the set (5), resulting in a mean value of 70 degrees.

Median

Another measure of central tendency used in climate statistics is the median, which serves to identify the middle value in a set of values. To find the median, we first organize the data points in ascending order, from the smallest to the largest. Once the data set is ordered, if there is an odd number of values, the median is the middle value. For instance, in the temperatures measured above, we first sort and get 66, 67, 69, 74, 74. Since there are five numbers, the third number (the middle one) is the median: 69.

The “middle” number is obvious when we have an odd number of data points (like 3, 5, or 173), but what do we do with an even number of data points where there are effectively “two” middle numbers? In this case, finding the median involves an additional step. It is computed by averaging the two middlemost values of the ordered sequence. Let’s say we add an additional measurement to our temperature above. Now we are at 69, 74, 67, 74, 66, 75. We sort to get 66, 67, 69, 74, 74, 75. Then, we take the two middle numbers, 69 and 74 in this case, and average them (71.5). Here, the median isn’t even a number in our dataset, but splits the difference between two of them!

The median can be more useful than the mean in the cases of distributions that have outliers. An outlier is an extreme value that extends far away from many of the other numbers in the dataset. In our first example, let’s replace the number 66 with 21 – a very cold day! Now the mean is 69+74+67+74+21=305, and 305 divided by 5 tells us our mean is 61. Just that single cold day, which represents an outlier to the rest of the dataset drags our mean down 9 degrees! However, note that our median stays the same, since the ordering becomes 21, 67, 69, 74, 74, and 69 remains the central value.

Mode

Our final measure of central tendency is the mode – defined as the value that appears most frequently in a given data set. Identifying the mode helps climate scientists recognize the most occurring or typical value in the dataset. For example, using our temperature dataset from before (66, 67, 69, 74, 74), the mode would be 74, as it appears more frequently than the other temperatures. If no number repeats, the set is said to have no mode, and in cases where two or more values occur with the same highest frequency, the dataset is considered to be multimodal, implying it has multiple modes (bimodal means two modes, trimodal means three modes, etc.).

Understanding and identifying the mode is significant as it gives insight into the most common or frequent occurrence in a dataset, offering a different perspective compared to the mean and median. Unlike the mean, the mode is not affected by extreme values or outliers, and, unlike the median, it is not necessarily positioned at the center of the ordered dataset.

Outliers

Speaking of outliers, we should talk about distributions. A distribution in climate science represents the set of all possible values or outcomes of a variable, such as temperature or precipitation, and defines how frequently each value occurs within a given dataset. Essentially, it provides a summary of the probability of occurrences of different possible outcomes in an experiment, offering a depiction of the variability or spread of a particular quantity. These are commonly shown graphically with a line or bar graph, with the horizontal x-axis containing the values of the variable and the vertical y-axis telling us something about their frequency of occurrence.

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There are three types of distributions we’ll primarily use this semester.

Uniform Distribution:

In a Uniform Distribution, every value in the dataset has an equal likelihood of occurring. On a graph, this is represented graphically as a flat, horizontal line. This distribution is characterized by the minimum and maximum values of the distribution. The probability of any given value occurring within this range is constant, and values outside of this range have a probability of zero. For example, in our dice example above, when rolling a single die, the odds of rolling one single number are equally likely, between 1 and 6, demonstrating a uniform distribution.

Eagle-eyed observers might take note of the statistics in the top-right corner of the figure, which show a mean of 3.5 (as expected), a median of 4, and a mode of 5. Wait—those bars look flat, so how is the mode 5!? These small deviations from perfect uniformity arise because the distribution is based on a finite (though large) sample of rolls rather than an infinite one. To make this clearer, I have added the number of rolls for each outcome above the bars, highlighting these slight differences even though the overall distribution appears very flat. With truly infinite rolls, the median would settle exactly at 3.5, and no single number would stand out as the mode.

Normal Distribution:

Uniform distributions are simple, but quite rare in nature. A common distribution seen in climate science – and quite prevalent across natural phenomena -- is the Normal Distribution, sometimes known as a Gaussian Distribution. This distribution follows a “bell-shaped” curve, with values closer to the central tendency (the mean and median!) being far more likely to occur than values in the “tails” of the distribution (very warm or very cold temperatures, for example). We also can briefly introduce the idea of “standard deviation” which is a measure of how much variation exists within a dataset. It is usually denoted by the Greek letter sigma. You may recognize these values, since some professors will “curve” their class grades based on this distribution (but not this one!). Small standard deviations tell us the distribution is very narrow, and all values tend to fall very close to one another. Large standard deviations tell us the opposite. If you are in a class where the mean on an exam was 81, and the standard deviation was 2, this means a *lot* of people scored very close to 81! On the other hand, if you have a mean of 81, but a standard deviation of 15, it means the distribution was far more spread out. In a symmetric normal distribution, about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Sometimes, you will hear people refer to something as a “three-sigma” outcome – this comes up in all walks of life, not just climate science. It means that it’s an event that falls at or outside the three standard deviation limit. If 99.7% of events fall within 3 standard deviations, it means a three-sigma events has less than a 0.3% chance of occurring in your distribution. Let's consider the IQ scores of a population, if IQ scores follow a normal distribution with a mean of 100 and a standard deviation of 15. An IQ score of 145 or higher is in the top 0.3% of the population, as it represents individuals with exceptionally high intelligence – this individual could be considered to score at a three sigma level (or three sigmas above the mean).

In the case of this symmetric distribution, the mean, median, and mode of a normal distribution are all equal and located smack-dab at the center. The standard deviation controls the spread or width of the distribution: a smaller standard deviation results in a narrower peak, while a larger one leads to a wider curve.

Skewed Distribution:

A Skewed Distribution can be thought of as a cousin of the normal distribution. In climate science, it typically follows something that has a lopsided bell curve. In other words, it is asymmetrical and does not exhibit mirror-image symmetry around the central value. Distributions can be either positively skewed or negatively skewed. In a positively skewed distribution, the tail on the right-hand side is longer than the left-hand side, indicating that most of the values are concentrated on the left, with a few extreme values to the right. Conversely, a negatively skewed distribution has a longer tail on the left-hand side. Unlike our normal distribution, the mean, median, and mode in a skewed distribution are not equal. Typically, in a positively skewed distribution, the mean is greater than the median, which is greater than the mode, and in a negatively skewed distribution, the mode is greater than the median, which is greater than the mean.

A variable that is positively skewed in climate science is the precipitation distribution at a particular point. Think about living in central Pennsylvania. Most of the time there is very little rain, or no rain at all. When it does rain, it’s generally more of a nuisance than anything. But occasionally, it can rain very hard or for very long periods of time. These events are rare and are outliers. Check out the graph below, the “tail” stretches much longer to the right-hand side, indicating a positively skewed distribution.

Now we roll 4 weighted dice one million times and add up what their faces show. These dice have small pieces of metal inside of them so that low numbers come up more often than high numbers. This results in a skewed distribution where combinations of lower numbers happen more frequently than higher numbers. This is an example of a positively skewed distribution, since the “tail” is longer on the right-hand side of the measures of central tendency. Also note that the mean is greater than the median, which is greater than the mode.

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Time series

We are typically interested how some component evolves over time and if it is changing, what is causing it to change. A time series is a fundamental data analysis approach used to gain insights into the behavior of climatic variables over some period of time. You have almost certainly come across these in other aspects of your day-to-day life, such as the price movement of the stock market or how Major League Baseball players are dealing with new rules.

In climate science, this typically involves taking some measurement or variable and quantifying how it changes over some time. This time period can be somewhat arbitrary – one could look at the annual climate of a particular region (the time period is one year long) or multiple millennia. This type of analysis permits a clear representation of trends, patterns, and fluctuations of the variable over time. By enabling the visualization of data points sequentially over time, time series graphics facilitate the identification of any consistent patterns, cyclic behaviors, abrupt changes, or otherwise weird behavior in the data.

For example, a climatological time series chart below allows us to see the evolution of temperature in State College, PA. It tells us much of what we already know, that the warmest months are in July and August on average. But did you know that the month with the most precipitation is May, followed closely by September? We’ll talk more about how the circulation of the atmosphere contributes to different time series in the climate later in the semester.

Trend

Over longer periods of time (say years to decades and beyond), we might be very interested in how a particular variable is changing over time. In combination with a time series, a trend line helps us see the overall direction in which a set of data points is moving over time. Imagine you have a graph where each dot represents a data point, like the temperature measured at different times throughout the year. These dots might seem scattered and chaotic, making it difficult to discern any pattern immediately. This is where a trend line comes into play. By drawing a straight line that best fits through these scattered dots, the trend line simplifies the complexity of the data, offering a clear, straightforward visual of whether the overall trend as a function of time is upwards, downwards, or relatively flat. It's like connecting the dots in a way that reveals the bigger picture, helping us to see beyond the short-term fluctuations and understand the broader, long-term pattern.

Temperature is actually a great example to use in a climate course. See the chart below which shows a time series of surface temperature averaged over the entire United States. Each point represents a different year. The jagged line bouncing up and down shows that there is a lot of year-to-year variability in the data. This type of jumping is usually associated with something known as “internal variability,” which we’ll talk about later in the class. The blue line represents a line (here, a linear regression) that shows the underlying trend in the data. Exactly how it’s calculated isn’t something you’ll need to do, but just notice that once you overlay this “best fit” line, you see the long-term trend from cooler temperatures to higher temperatures as you move from left to right (forward in time!). This represents a positive trend- temperatures in the United States have slowly but steadily increased over the past century.

Anomaly

It is very useful to understand the underlying distribution of variables, such as temperature or precipitation, but there are many instances where we want to know how far something deviates from its mean climatology. Having 6 inches of snow in northern Maine in December may barely induce school delays, but 6 inches of snow in Atlanta can gridlock traffic for days, even long after the snow has melted. To understand how a particular variable varies from a baseline state, climate scientists commonly calculate something known as an anomaly. An anomaly refers to the deviation of a particular variable from its long-term average over a specific time period. To calculate a climate anomaly, scientists first establish a baseline or reference period, often a time period spanning multiple decades, to represent typical or "normal" conditions for a specific location or region. This baseline is essential because it provides a standard against which current or future climate data can be compared. Once the baseline is established, the anomaly is simply the difference between the observed value and this baseline value. This difference, typically expressed as a numerical value or anomaly, indicates whether the recent climate conditions were warmer, cooler, wetter, drier, or otherwise different from the long-term average. As a simple example, if the temperature in State College on July 4th has averaged 82F over the past 30 years, then having an Independence Day holiday with a 99F temperature represents a +17F anomaly. Anomalies can be displayed in a variety of different ways. A spatial map of the air temperature anomalies during the 2021 Pacific Northwest heat wave is shown below. All of the red areas show where air temperatures climbed more than 27°F (15°C) higher than the 2014-2020 average for the same day. In other words, if Seattle normally was 80F on that day, they were actually observing temperatures of 107F!

Climate anomalies play a pivotal role in climate science for a couple of reasons. First, they allow scientists to identify and quantify variations and trends in climate data. By comparing recent climate conditions to the long-term average, researchers can detect patterns of change, such as long-term warming trends, shifts in precipitation patterns, or the occurrence of extreme weather events. These anomalies provide valuable insights into how our climate is evolving over time. Second, climate anomalies are crucial for assessing the impacts of climate change. By calculating anomalies, scientists can determine whether specific regions are experiencing changes that are outside the bounds of natural variability. This information is essential for understanding the extent to which human activities, such as greenhouse gas emissions, are influencing our climate. Identifying areas where anomalies are consistently occurring helps policymakers and communities prepare for and adapt to changing climate conditions, whether that means addressing the risks associated with sea-level rise, altered precipitation patterns, or more frequent heatwaves. Understanding how things evolve in ways that are different from the baseline we are accustomed to will be a common theme throughout the remainder of this course.

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Summary

• Climate is the long-term average of weather patterns in a specific region, encompassing various components like the atmosphere, hydrosphere, cryosphere, lithosphere, and biosphere, which are all interconnected.
• Understanding climate is crucial as it affects agriculture, water availability, ecosystems, energy usage, air quality, and the resilience of coastal regions, influencing many aspects of life and society.
• Scientists study climate through observations, models, and experiments, each providing different insights that collectively enhance our understanding of the Earth's climate system.
• Analyzing climate data involves understanding distributions, trends, and anomalies to identify changes over time, which is essential for assessing the impacts of climate change and preparing for future challenges.
• A basic understanding of some statistics commonly used by scientists will come in handy for the rest of this class!

Now that we have an understanding as to what climate actually is, it begs the question, "how do we actually observe it?" Sounds like a good topic for the next lesson!

Before completing the assessments, take a few minutes to take the quiz below.