What Is The Normal Distribution And The Empirical Rule Example Explained, 68 95 and 97.7 rule

Summary

In this video by 'Whats Up Dude', the concept of normal distribution and the empirical rule are clearly explained. A normal distribution is a probability distribution that is symmetric and bell-shaped, determined by its mean and standard deviation. Key properties include its mean, median, and mode being equal; the curve is continuous and never touches the x-axis. Importantly, the empirical rule states that approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. This video breaks down these concepts in a simple manner.

Highlights

• Normal distribution is defined by its mean and standard deviation 🔢.
• Empirical rule helps understand data spread across standard deviations 🚀.
• Visual demonstration of symmetrical data curve 🎥.
• Breakdown of area percentages and how they are calculated 📏.
• Key points to remember: total area equals 1, empirical rule application 💡.

Key Takeaways

• Normal distribution is symmetric and bell-shaped 🎨.
• Its mean, median, and mode are centrally located 🔍.
• The empirical rule: 68% (1 SD), 95% (2 SDs), 99.7% (3 SDs) 📊.
• Continuous curve that never touches the x-axis ➖.
• Useful breakdown of areas under the curve 📐.

Overview

The video begins by introducing the concept of normal distribution, emphasizing its bell shape and symmetry. This distribution is crucial for understanding data spreads as it effectively utilizes mean and standard deviation to stabilize its form. The visual representation with varying peaks helps in grasping the disparity created by different standard deviations.

Key properties of normal distribution include its peak points where the mean, median, and mode converge, making it symmetric. The continuity of the curve is underscored by its never-ending form that doesn't touch the x-axis, a concept cemented by calculus. Highlighting these traits signals the importance of understanding these definitions for practical applications.

Finally, the video dives into the empirical rule, a useful tool in statistics that clarifies the spread of data across standard deviations. The rule provides percentages that translate to real-world applications, allowing for a better grasp of data distribution. This part of the video solidifies understanding by simplifying intricate calculations into digestible information.

Chapters

• 00:00 - 00:30: Introduction to Normal Distribution A normal distribution is a probability distribution for a continuous random variable characterized by its bell shape and symmetry. Its appearance and position depend on the mean and standard deviation. The y-axis is often excluded because the total area under the curve is more significant. For example, two normal distributions can have the same mean but different spreads, determined by their standard deviation.
• 00:30 - 01:00: Properties of Normal Distribution The chapter titled 'Properties of Normal Distribution' discusses key characteristics of the normal distribution. Firstly, it states that in a normal distribution, the mean, median, and mode are equal and located at the center. Secondly, the curve is bell-shaped and symmetric about the mean, implying that if a line is drawn down the center, the shape and area on both sides are identical. Additionally, the chapter notes that the curve is continuous, extending infinitely without touching the x-axis. Lastly, it emphasizes that the total area under the normal distribution curve equals 1. These properties are crucial for understanding the distribution's behavior and characteristics.
• 01:00 - 02:00: The Empirical Rule Explained This chapter explains the Empirical Rule as it applies to normal distribution curves. It highlights that approximately 68% of the area under the curve lies within one standard deviation of the mean. This implies that 68% of data points are expected to fall between one standard deviation below and above the mean. The video refrains from delving into the calculus proof, focusing instead on the key concept of the rule and its significance in understanding data distribution.
• 02:00 - 03:00: Breaking Down the Empirical Rule The chapter 'Breaking Down the Empirical Rule' explains the distribution of data in a normal distribution based on the empirical rule. It covers how 68% of the data falls within one standard deviation from the mean, with 34% on each side, and 95% of the data falls within two standard deviations, with 47.5% on each side. The chapter emphasizes the symmetry of the normal distribution and how areas under the curve are divided equally on either side of the mean.
• 03:00 - 03:30: Conclusion and Summary The chapter discusses the concept of standard deviations and the distribution of data within a statistical curve. It highlights the empirical rule that approximately 99.7% of data will fall within three standard deviations from the mean. By dividing this percentage, it explains how 49.85% of data lies on either side of the mean. The chapter further explores these percentages and the symmetry of the distribution around the mean. The discussion focuses on calculating the area under the curve, especially examining the right side of the distribution, detailing that 47.5% of data lies from the mean up to two standard deviations.

What Is The Normal Distribution And The Empirical Rule Example Explained, 68 95 and 97.7 rule Transcription

• 00:00 - 00:30 A normal distribution is a probability  distribution for a continuous random variable   that has a bell shape and is symmetric, like you  see here. When it is drawn the y-axis is rarely   included, because the area under the distribution  curve is used more than the values on the y-axis.   Its shape, how high it is so to speak, and its  location depend on two things, the mean, and   the standard deviation. For instance here are 2  normal distributions graphed together, the red one   has a higher peak and the green one is more spread  out. They both have the same mean but the green one
• 00:30 - 01:00 has a larger standard deviation, making it more  spread out. There are some important properties   of a normal distribution. Number 1 is that the  mean, median, and mode are equal, and are located   at the center of the distribution. Number 2, its  curve is bell-shaped and symmetric about the mean,   so if we draw a line down the middle, the shape and  area is the same on both sides of the line. Number   3 is that the curve is continuous, meaning that  it goes on forever and never touches the x-axis.   Number 4 is that the total area under the  normal distribution curve is equal to 1. This
• 01:00 - 01:30 may seem wrong as a curve never touches the x  axis, but it can be proven using calculus, which   we're not going to do in this video. Number 5  is that the empirical rule applies to a normal   distribution curve and it states that about  68% of the area under the curve, lies within 1   standard deviation of the mean. Within 1 standard  deviation means 1 standard deviation to the left,   and one to the right, so about 68% of the area lies  between -1 and positive 1 standard   deviation. Since we know the shapes and areas are  the same on both sides of the mean, we can divide
• 01:30 - 02:00 68% in half to get 34% on both sides. The empirical  rule also states that about 95% of the area under   the curve, lies within 2 standard deviations of  the mean, within 2 standard deviations means 2   standard deviations to the left and 2 to the  right. Since we know the shapes and areas are the   same on both sides of the mean, we can divide 95%  in half to get 47.5% on both sides. The empirical
• 02:00 - 02:30 rule also states at about 99.7% of the area under the curve lies   within 3 standard deviations of the mean.  Again within 3 standard deviations means   3 standard deviations to the left and 3  to the right. Since we know the shapes and areas are   the same on both sides of the mean, we can divide  99.7% in half to get 49.85%  on both sides. And we can break these areas down  further by doing some subtraction. Let's look at   just the right side for now. We have 47.5%,  from the mean, to 2 standard
• 02:30 - 03:00 deviations, and 34% from the mean  to 1 standard deviation. If we subtract 34 from   47.5, we get 13.5,  and that would be the percentage of area   between 1 and 2 standard deviations. We can  do the same for the area between 3 standard   deviations and 2 standard deviations, 49.85 minus 47.5   equals 2.35. So 2.35%  of the area lies between 2 and   3 standard deviations. Again, since we know  the shapes and areas are the same on both sides
• 03:00 - 03:30 of the mean, we can apply these percentages to the  left side of the mean. The 2 biggest things to   remember about the normal distribution curve is  that the area under the curve is equal to 1, and   the empirical rule of 68%, 95% and 99.7%   in regards to the mean and standard deviation.  Both of these concepts are extremely important.   All right my friends, that be the basics on  the normal distribution. Hopefully you got   something out of this video, I do have more videos  right there for you, till next time, I am outta here.