# Shapes of distributions

##### Intros

###### Lessons

##### Examples

###### Lessons

**Exploring Different Types of Distributions:**

In Mrs. Katsufrakis' class the scores for a test are given in the graph below:

What is the average score?

i. 90%

ii. 80%

iii. 70%

- A bank assures you that one of their tellers will be of assistance to you in 5 minutes or less. Further the bank tells you that the teller will become available equally likely at any time throughout those first 5 minutes. What does the distribution of this wait time look like?
- An IQ test has an average score of 100 and has more data clumped closer to the mean than further away. What might be the distribution of an IQ test?
**Clusters, Gaps and Peaks**

The finishing times of a marathon are given in the following distribution below:

Signify where all the clusters, gaps and peaks are.

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###### Topic Notes

## Introduction to Shapes of Distributions

Welcome to our exploration of the fascinating world of distribution shapes in statistics! Understanding the shape of distribution is crucial for interpreting data and making informed decisions. In our introduction video, we'll dive into this essential concept, breaking it down in a way that's easy to grasp. You'll learn about various data distribution patterns, from symmetrical bell curves to skewed and bimodal distributions. We'll discuss how these shapes can reveal important insights about your dataset, helping you identify trends, outliers, and potential data collection issues. By the end of this lesson, you'll have a solid foundation in recognizing and interpreting different shapes of distributions. This knowledge will be invaluable as you progress in your statistical journey, enabling you to analyze data more effectively and draw meaningful conclusions. So, let's get started on this exciting adventure into the world of distribution shapes!

## Understanding Distributions and Their Types

Distributions play a crucial role in statistics, serving as powerful tools for organizing, summarizing, and analyzing data. At its core, a distribution is a way of representing how data is spread out or arranged. Understanding distributions is essential for anyone working with data, as they provide valuable insights into patterns, trends, and characteristics of datasets.

There are two main types of distributions that statisticians and data analysts commonly work with: frequency distributions and probability distributions. Let's explore each of these in detail to grasp their significance and applications.

### Frequency Distributions

A frequency distribution is a way of organizing and summarizing data by showing how often different values or categories occur within a dataset. This type of distribution is particularly useful for visualizing and analyzing large sets of data.

For example, imagine a teacher collecting test scores for a class of 30 students. The scores range from 60 to 100. A frequency distribution might look like this:

- 60-69: 3 students
- 70-79: 7 students
- 80-89: 12 students
- 90-100: 8 students

This frequency distribution quickly shows that most students scored between 80 and 89, providing a clear picture of the class's performance. Frequency distributions can be represented in various formats, including tables, histograms, and bar charts, making them versatile tools for data visualization.

### Probability Distributions

Probability distributions, on the other hand, describe the likelihood of different outcomes in a random event or experiment. These distributions are fundamental to probability theory and statistical inference.

A classic example of a probability distribution is the roll of a fair six-sided die. The probability distribution for this scenario would be:

- 1: 1/6 probability
- 2: 1/6 probability
- 3: 1/6 probability
- 4: 1/6 probability
- 5: 1/6 probability
- 6: 1/6 probability

This distribution shows that each outcome has an equal probability of occurring. Probability distributions can be discrete (like the die example) or continuous (such as the normal distribution, which is often used to model natural phenomena).

### How Distributions Represent Data

Distributions are powerful because they provide a mathematical framework for representing and analyzing data. They allow us to:

- Summarize large datasets concisely
- Identify patterns and trends in data
- Make predictions and inferences about populations based on samples
- Compare different datasets or populations
- Understand the variability and spread of data

When we talk about a "data distribution math definition," we're essentially describing the mathematical function or rule that defines how the data is spread out. This could be a simple list of values and their frequencies (as in a frequency distribution) or a more complex mathematical function (as in many probability distributions).

### Types of Graph Distributions

Distributions can be visualized using various types of graphs, each suited to different kinds of data and analysis needs. Some common types include:

- Histograms: Ideal for continuous data, showing frequency of data falling within specific intervals
- Bar charts: Great for categorical data, displaying frequency or count for each category
- Line graphs: Useful for showing trends over time or continuous variables
- Box plots: Excellent for displaying the spread and skewness of data
- Scatter plots: Perfect for showing relationships between two variables

Understanding these different types of graph distributions is crucial for effectively communicating

## Symmetric Distributions and Bell-Shaped Curves

Symmetric distributions play a crucial role in statistics, with the bell-shaped or normal distribution being one of the most prominent examples. These distributions are characterized by their symmetry around a central point, creating a mirror image on either side. The bell-shaped histogram, also known as a mound-shaped distribution, is particularly common and important in statistical analysis.

A bell curve shape is easily recognizable by its distinctive appearance. Picture a smooth, rounded curve that resembles the shape of a bell or a mountain. This curve is highest in the middle and gradually tapers off on both sides, creating two "tails" that extend outward. The symmetry of this shape is key to understanding its properties and applications in various fields.

One of the most important characteristics of a bell-shaped curve is its mean, which is located at the center of the distribution. In a perfectly symmetric distribution, the mean, median, and mode all coincide at this central point. This alignment of measures of central tendency is a hallmark of the normal distribution and contributes to its usefulness in statistical analysis.

Another crucial aspect of the bell curve shape is the concentration of data around the mean. In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This predictable pattern of data distribution is known as the empirical rule or the 68-95-99.7 rule, and it's a fundamental concept in statistics.

The prevalence of bell-shaped histograms in statistics is not coincidental. Many natural phenomena and human-related measurements tend to follow this distribution. For example, human height, blood pressure readings, and exam scores often exhibit a bell-shaped pattern when plotted. This ubiquity is partly explained by the Central Limit Theorem, which states that the sum of a large number of independent random variables tends to follow a normal distribution, regardless of the underlying distribution of those variables.

To illustrate the concept of a symmetric distribution example, consider a class of 100 students taking a standardized test. If we plot their scores on a histogram, we might find that most students score near the average, with fewer students achieving very high or very low scores. This creates a bell-shaped curve, with the peak representing the most common scores and the tails representing the less frequent extreme scores.

Another example of a mound-shaped distribution can be observed in manufacturing. If we measure the diameter of machine-produced bolts, we'd likely find that most bolts are very close to the target size, with fewer bolts being significantly larger or smaller. When plotted, this data would form a symmetric, bell-shaped curve centered around the target diameter.

The normal distribution's importance extends beyond its frequent occurrence in nature. It serves as a foundation for many statistical techniques and hypothesis tests. Its well-defined properties allow researchers to make predictions and draw inferences about populations based on sample data. For instance, in quality control, manufacturers can use the normal distribution to set acceptable limits for product variations and identify when a process is out of control.

While the normal distribution is the most well-known symmetric distribution, it's important to note that not all symmetric distributions are bell-shaped. Other examples include the uniform distribution (where all values within a range are equally likely) and the t-distribution (which resembles the normal distribution but has heavier tails). However, the bell-shaped curve remains the most commonly encountered and studied form of symmetric distribution in statistics.

Understanding symmetric distributions, particularly the bell-shaped or normal distribution, is essential for anyone working with data analysis, research, or any field that relies on statistical inference. Its predictable properties and widespread applicability make it a powerful tool for understanding variability in data and making informed decisions based on statistical analysis. Whether you're a student, researcher, or professional, recognizing and interpreting bell-shaped histograms and other symmetric distributions will enhance your ability to analyze and draw meaningful conclusions from data.

## Skewed and Uniform Distributions

Understanding different types of distributions is crucial in statistics and data analysis. In this section, we'll explore skewed distributions, uniform distributions, and compare them to symmetric distributions, providing real-world examples to enhance comprehension.

Skewed distributions are characterized by an asymmetrical shape, where the data is concentrated more on one side than the other. There are two types of skewed distributions: left-skewed and right-skewed.

Left-skewed distributions, also known as negatively skewed, have a longer tail on the left side of the graph. The majority of the data points are concentrated on the right side, with the mean typically being less than the median. An example of a left-skewed distribution is the age at which people retire. Most people retire around a certain age, but some retire much earlier, creating a tail on the left side of the distribution.

Right-skewed distributions, or positively skewed, have a longer tail on the right side of the graph. The bulk of the data is concentrated on the left side, with the mean usually being greater than the median. Income distribution in many countries is a classic example of a right-skewed distribution. Most people earn average or below-average incomes, while a smaller number of high earners create a long tail on the right side.

Uniform distributions, on the other hand, have a completely different shape. In a uniform distribution, all outcomes or values within a given range have an equal probability of occurring. Graphically, this results in a rectangular shape, often referred to as a "u shaped distribution" or an "approximately uniform histogram."

The key characteristics of a uniform distribution include:

- Equal probability for all values within the range
- A constant probability density function
- The mean, median, and mode are all equal and located at the center of the distribution

Real-world examples of uniform distributions include:

- The probability of rolling any number on a fair six-sided die
- The distribution of birthdays throughout the year (assuming no seasonal variations)
- The arrival times of customers at a store during a specific hour

When comparing uniform distributions to symmetric distributions, it's important to note that while all uniform distributions are symmetric, not all symmetric distributions are uniform. Symmetric distributions have a mirror image on either side of the central point, with the mean, median, and mode all being equal. The most well-known symmetric distribution is the normal distribution, also called the Gaussian distribution or bell curve.

The key differences between uniform and symmetric distributions are:

- Shape: Uniform distributions have a flat, rectangular shape, while symmetric distributions can have various shapes, such as the bell curve of a normal distribution.
- Probability density: In a uniform distribution, the probability density is constant across the range, whereas in other symmetric distributions, it can vary.
- Tails: Uniform distributions have abrupt cutoffs at the ends of their range, while many symmetric distributions, like the normal distribution, have tails that extend infinitely in both directions.

Understanding the differences between skewed, uniform, and symmetric distributions is crucial for accurately interpreting data and making informed decisions. Skewed distributions often represent real-world phenomena where extreme values are more common in one direction, such as wealth distribution or reaction times. Uniform distributions are less common in nature but can be useful in modeling certain random processes or in theoretical probability problems.

When analyzing data, it's important to consider the shape of the distribution, as it can significantly impact the choice of statistical methods and the interpretation of results. For instance, many statistical tests assume a normal distribution, and using them on heavily skewed data without appropriate transformations can lead to incorrect conclusions.

In conclusion, recognizing and understanding different distribution shapes whether skewed, uniform, or symmetric is a fundamental skill in data analysis and statistics. By being able to identify these patterns and their real-world applications, analysts can better interpret data, choose appropriate statistical methods,

## Identifying Clusters, Gaps, and Peaks in Distributions

Understanding the shape of a distribution is crucial for data analysis and interpretation. Three key features that often appear in distributions are clusters, gaps, and peaks. These characteristics provide valuable insights into the underlying patterns and structure of data. In this section, we'll explore what these features are, how to identify them, and their impact on data interpretation.

Clusters in a distribution refer to groups of data points that are closely packed together. These concentrations of data indicate that certain values or ranges are more common within the dataset. To visualize clusters, imagine a histogram where several bars are taller and closer together in specific areas. Clusters can reveal subgroups or patterns within the data that might not be immediately apparent from summary statistics alone.

Gaps in a distribution are areas where there are few or no data points. These empty spaces or low-frequency regions can be just as informative as clusters. Gaps may indicate natural divisions in the data, outliers, or even data collection issues. In a histogram, gaps appear as areas with very short or nonexistent bars between more populated regions. Identifying gaps can help in understanding the overall structure of the data and potential underlying factors influencing the distribution.

Peaks in a distribution are the highest points or local maxima in the data's frequency. They represent the most common values or ranges within the dataset. In a histogram, peaks appear as tall bars that stand out from their neighbors. A distribution can have one peak (unimodal), two peaks (bimodal), or multiple peaks (multimodal). The number and location of peaks provide crucial information about the central tendencies and potential subgroups within the data.

To identify these features in various distribution shapes, it's essential to examine the data visually and statistically. Histograms are particularly useful for this purpose, as they provide a clear representation of the data's frequency distribution. When analyzing a histogram, look for areas where bars are clustered together (clusters), regions with very low or no bars (gaps), and the tallest bars or groups of tall bars (peaks).

In a normal distribution, you'll typically see a single, symmetrical peak in the center, with data gradually tapering off on both sides. This bell-shaped curve doesn't usually exhibit distinct clusters or gaps. However, in real-world data, distributions often deviate from this ideal shape, and that's where identifying clusters, gaps, and peaks becomes crucial.

For example, a bimodal distribution will show two distinct peaks, which could indicate the presence of two subgroups within the data. This might be seen in a height distribution of a mixed-gender population, where one peak represents the average height for women and another for men. In this case, the gap between the peaks is also informative, showing the separation between these two subgroups.

A skewed distribution might have a long tail on one side, with the peak shifted away from the center. This could reveal clusters of data points at one end of the distribution, with a gradual tapering off towards the other end. Such a shape might be observed in income distributions, where there's a cluster of lower incomes, a peak representing the median, and a long tail extending towards higher incomes.

Identifying the shape of the distribution through these features significantly affects data interpretation. Clusters can indicate subgroups or patterns that warrant further investigation. For instance, in customer segmentation, clusters might reveal distinct groups of customers with similar behaviors or preferences. Gaps might signify natural breakpoints in the data or potential data quality issues that need addressing. Peaks provide insights into the most common values or ranges, which is crucial for understanding central tendencies and making predictions.

Moreover, the presence of multiple peaks or unexpected gaps can challenge assumptions about the data's underlying distribution. This is particularly important when choosing statistical methods, as many techniques assume a normal distribution. Recognizing non-normal features in your data can guide you towards more appropriate analytical approaches.

In conclusion, clusters, gaps, and peaks are essential features for understanding the shape of distributions. By carefully examining these characteristics, analysts can gain deeper insights into their data, identify potential subgroups or anomalies, and make more informed decisions about further analysis and interpretation. Whether you're working with simple histograms or more complex multivariate distributions, the ability to recognize and interpret these features is a valuable skill in data analysis and statistics.

## Interpreting Different Shapes of Distributions

Understanding how to describe the shape of a distribution is crucial in statistical analysis. The shape of a distribution provides valuable insights into the underlying data and helps determine appropriate statistical measures. In this guide, we'll explore various distribution shapes, their implications, and how to interpret them effectively.

### Common Shapes of Distributions

1. Normal Distribution (Bell Curve): This symmetrical, bell-shaped curve is characterized by its central peak and tapering tails. It's often found in natural phenomena and is crucial in many statistical analyses.

2. Skewed Distributions: These can be either positively (right) or negatively (left) skewed. In a positively skewed distribution, the tail extends to the right, while in a negatively skewed distribution, it extends to the left.

3. Uniform Distribution: This shape shows equal probability across all values, resulting in a rectangular shape.

4. Bimodal Distribution: Characterized by two distinct peaks, this shape suggests two different subgroups within the data.

5. Multimodal Distribution: Similar to bimodal, but with more than two peaks, indicating multiple subgroups.

### Interpreting Distribution Shapes

Normal Distribution: When data follows a normal distribution, it suggests that the majority of values cluster around the mean, with fewer extreme values. This shape is ideal for many statistical tests and allows for easy prediction of probabilities.

Skewed Distributions: A skewed shape indicates that the data is not symmetrically distributed around the mean. In positively skewed distributions, the mean is typically greater than the median, while in negatively skewed distributions, the mean is less than the median. Skewness can affect the choice of statistical measures and tests.

Uniform Distribution: This shape suggests that all outcomes are equally likely. It's less common in natural phenomena but can be found in certain random processes or artificially created scenarios.

Bimodal and Multimodal Distributions: These shapes indicate the presence of distinct subgroups within the data. They often suggest that the data comes from two or more different populations or processes.

### Choosing Appropriate Statistical Measures

The shape of a distribution greatly influences which statistical measures are most appropriate:

1. For Normal Distributions: Mean and standard deviation are suitable measures of central tendency and spread.

2. For Skewed Distributions: Median and interquartile range are often more appropriate, as they are less affected by extreme values.

3. For Bimodal or Multimodal Distributions: It may be necessary to analyze each mode separately or consider using non-parametric tests.

### Practical Examples

Example 1: Income Distribution Income data often shows a positively skewed distribution. Here, using the median income is more representative than the mean, which can be inflated by a few high earners.

Example 2: Test Scores In a normal distribution of test scores, the mean score is a good representation of the average student performance. Standard deviation can effectively describe the spread of scores.

Example 3: Age Distribution in a Community A bimodal distribution of ages in a community might indicate two distinct groups, such as young families and retirees. Analyzing these groups separately may provide more meaningful insights.

### How to Find the Shape of Distribution

1. Visual Inspection: Create histograms or box plots to visually assess the shape.

2. Quantitative Measures: Use skewness and kurtosis statistics to quantify the shape.

3. Statistical Tests: Employ tests like the Shapiro-Wilk test to check for normality.

4. Q-Q Plots: These plots compare the distribution of your data to a theoretical distribution (often normal).

### Conclusion

Understanding how to describe the shape of a distribution is fundamental in statistical analysis

## Conclusion: The Importance of Understanding Distribution Shapes

Understanding the shapes of distributions is crucial in statistical analysis. As demonstrated in the introduction video, distribution shapes provide valuable insights into data patterns and characteristics. Key points include recognizing symmetric versus skewed distributions, identifying unimodal and multimodal patterns, and understanding the significance of outliers. These concepts are fundamental to interpreting data accurately and making informed decisions. The shape of a distribution influences statistical measures and helps in selecting appropriate analytical techniques. By mastering these concepts, you'll enhance your ability to analyze and interpret data effectively. We encourage you to explore this topic further, delving into specific applications in various fields such as economics, biology, and social sciences. Remember, the shape of a distribution is more than just a visual representation; it's a powerful tool in statistical inference. Continue your learning journey by practicing with real-world datasets and exploring advanced statistical concepts. Your understanding of distribution shapes will be invaluable in your future statistical endeavors.

### Shapes of Distributions

All the different ways to arrange data

i. Symmetric

ii. Skewed

iii. Bell-shaped

iv. Uniform

#### Step 1: Introduction to Distributions

In this section, we will discuss the shapes of distributions and provide a general overview of what distributions are. Distributions describe how data is spread out or distributed across different values. For example, if you were to measure the heights of students in a class, a distribution would show how many students fall into each height range. This can be visualized using a graph that displays where the data points are located.

#### Step 2: Types of Distributions

There are two main types of distributions: frequency distributions and probability distributions. A frequency distribution shows how often each value occurs in the data set. For instance, if you sampled 35 people and recorded their daily cholesterol intake, a frequency distribution would show how many people fall into each cholesterol range. On the other hand, a probability distribution is similar but always sums to 1. It shows the likelihood of each value occurring. For example, if you have a hand of 5 cards with 2 aces, 1 king, and 2 queens, the probability distribution would show the likelihood of drawing each card.

#### Step 3: Symmetric Distributions

A symmetric distribution is one where the data is evenly distributed around a central point. If you draw a vertical line through the center of the graph, the left and right sides will be mirror images of each other. This means that if you flip the graph along this line, the data points will match up perfectly. Symmetric distributions are common and can take various forms.

#### Step 4: Bell-shaped Distributions

A bell-shaped distribution, also known as a normal distribution or Gaussian distribution, is a specific type of symmetric distribution. It has a characteristic "bell" shape, with most of the data clustered around the mean (average) value. The peak of the bell curve represents the mean, and the data tapers off symmetrically on both sides. This type of distribution is very common in statistics and is often used in various mathematical and scientific applications.

#### Step 5: Skewed Distributions

Skewed distributions are those that are not symmetric. They can be skewed to the left or to the right. A left-skewed distribution has a longer tail on the left side, meaning that more data points are spread out towards the lower values. Conversely, a right-skewed distribution has a longer tail on the right side, with more data points spread out towards the higher values. Skewed distributions indicate that the data is not evenly distributed around the mean.

#### Step 6: Uniform Distributions

A uniform distribution is one where all values are equally likely to occur. This means that the data is spread out evenly across the range of possible values. In a graph, a uniform distribution appears as a flat, horizontal line, indicating that each value has the same frequency or probability. This type of distribution can occur in situations where each outcome is equally likely, such as rolling a fair die or drawing a card from a well-shuffled deck.

#### Conclusion

Understanding the different shapes of distributions is crucial for analyzing and interpreting data. Symmetric, bell-shaped, skewed, and uniform distributions each provide unique insights into how data is spread out. By recognizing these patterns, you can better understand the underlying characteristics of your data and make more informed decisions based on your analysis.

### FAQs

Here are some frequently asked questions about shapes of distributions:

#### 1. What are the main types of distribution shapes?

The main types of distribution shapes are symmetric (e.g., normal or bell-shaped), skewed (left or right), uniform, and bimodal or multimodal. Symmetric distributions have data evenly distributed around the center, skewed distributions have a tail extending to one side, uniform distributions have equal frequency across all values, and bimodal/multimodal distributions have two or more peaks.

#### 2. How do you describe the shape of a distribution?

To describe the shape of a distribution, consider its symmetry, skewness, number of peaks, and any unusual features. Key aspects to mention include whether it's symmetric or skewed, if it has one main peak (unimodal) or multiple peaks, and if there are any gaps or clusters in the data.

#### 3. What does a bell-shaped distribution indicate?

A bell-shaped distribution, also known as a normal distribution, indicates that data is symmetrically distributed around the mean. It suggests that most observations cluster around the average, with fewer observations towards the extremes. This shape is common in many natural and social phenomena.

#### 4. How does the shape of a distribution affect statistical measures?

The shape of a distribution affects which statistical measures are most appropriate. For symmetric distributions, mean and standard deviation are suitable. For skewed distributions, median and interquartile range are often better. The shape also influences the relationship between mean, median, and mode, and impacts the choice of statistical tests for analysis.

#### 5. What is a uniform distribution and when is it observed?

A uniform distribution is characterized by a constant probability for all values within a given range, resulting in a rectangular shape when graphed. It's observed in scenarios where all outcomes are equally likely, such as rolling a fair die or selecting a random number within a specified range. Uniform distributions are less common in natural phenomena but are important in probability theory and simulations.

### Prerequisite Topics

Understanding the shapes of distributions is a crucial concept in statistics, but to fully grasp its significance, it's essential to have a solid foundation in prerequisite topics. Two key areas that provide this foundation are the introduction to normal distribution and the central limit theorem.

The normal distribution is a fundamental concept in statistics, often referred to as the bell curve due to its characteristic shape. It serves as a cornerstone for understanding various shapes of distributions. By familiarizing yourself with the properties and characteristics of the normal distribution, you'll be better equipped to recognize and interpret other distribution shapes in real-world data sets.

Moreover, the normal distribution is not just a standalone concept; it plays a crucial role in many statistical analyses and probability calculations. Its symmetrical shape and well-defined properties make it a powerful tool for making inferences about populations based on sample data. As you delve deeper into the shapes of distributions, you'll find that many statistical techniques and assumptions are built upon the foundation of the normal distribution.

Equally important is the central limit theorem, which bridges the gap between theoretical concepts and practical applications in statistics. This theorem explains why many real-world phenomena tend to follow a normal distribution, even when the underlying data may not be normally distributed. Understanding the central limit theorem is crucial for grasping how different shapes of distributions can emerge from various sampling processes.

The central limit theorem also provides insights into why the normal distribution is so prevalent in statistical analysis. It helps explain how sample means from any distribution tend to approximate a normal distribution as the sample size increases. This knowledge is invaluable when interpreting and analyzing data in various fields, from social sciences to natural sciences and beyond.

By mastering these prerequisite topics, you'll develop a more intuitive understanding of how different factors influence the shapes of distributions. You'll be better prepared to recognize patterns, identify outliers, and make informed decisions based on statistical data. Furthermore, this knowledge will enhance your ability to choose appropriate statistical tests and models for different types of data distributions.

In conclusion, the journey to understanding the shapes of distributions is built upon a solid grasp of the normal distribution and the central limit theorem. These prerequisite topics provide the necessary context and theoretical framework to appreciate the nuances and implications of various distribution shapes encountered in statistical analysis. By investing time in these foundational concepts, you'll be well-equipped to tackle more advanced statistical topics and apply your knowledge to real-world problems with confidence.

__Frequency Distributions:__A graph representing the frequency of each outcome occurring.

__Probability Distributions:__A graph that provides the probability of each outcome occurring.

**The most common distribution shapes are:**

**Symmetric:**

**Bell-shaped:**

**Skewed to the left:**

**Skewed to the right:**

**Uniform:**

**Cluster:**

On a graph a cluster is when several data points lie close together. Typically this means that these data points have similar characteristics, which is also called things are

__homogeneous__(Things which are very diverse or dissimilar are called

__heterogeneous__)

**Gap:**

A gap is an interval which contains no data.

**Peak:**

A peak is the highest point of a data set.

remaining today

remaining today