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AP Statistics

Uses and Properties of Density Curves

Written by Prerit Jain

Updated on: 08 Dec 2023

Uses and Properties of Density Curves

Uses and Properties of Density Curves

A graph that displays probability is known as a density curve. All probabilities are represented by 100% of the curve’s area under the curve. You may alternatively state that the area is equal to 1 because probabilities are often expressed as decimals.

Using density curves to represent probability distributions

The area under the density curve has an area of 1. In other words, there is a 1 percent chance that a value will appear anywhere throughout the whole distribution curve. Finding the section of the area under the curve that corresponds to a narrower range of interest is necessary to determine the odds for that range.

In the above curve we can see that the probability that the value lies between 125 and 150 is as follows:

Finding the area of the shaded region will give us the probability of the value lying between 125 and 150P(125 < X < 150.

The median and quartiles are simple. The proportions of the total number of observations are represented by the areas under a density curve. The midpoint between two observations is known as the median.

The point where half the area under the curve is to its left and the other half is to its right is known as the equal-areas point, and it represents the median of a density curve. The area under the curve is divided into quarters by the quartiles.

The first quartile is to the left of one-fourth of the area under the curve, and the third quartile is to the left of three-fourths of the area. Any density curve’s median and quartiles may be roughly located by sight by splitting the curve’s under surface into four equal sections.

Properties of density curves

The shape of density curves

  • Symmetric density curves
    • A symmetric density curve is absolutely symmetric because density curves are idealized patterns. Therefore, the middle of a symmetric density curve is where the median lies.
  • Skewed density curve
    • Finding the point on a skewed curve with equal regions is more difficult. There are formulas that may be used to calculate the median of any density curve.
  • The density curves are always positive. For any feasible value of x, the probability density function is non-negative, or f(x) \ge 0. The density curve and horizontal X-axis have an area of one, or 1. The density function curve is continuous across the whole range because continuous random variables have this feature.
  • The area under the curve is always equated to 1. The probabilities of all the events occurring in the curve will always add up to one.

Creating and using density curves

Steps for creating a density curve from data

  1. For a data set, create the histogram where the x-axis shows the data values and the y-axis shows the frequency.
  2. We create a density curve to get the shape of the distribution.

We can interpret the density curves in different ways:

  1. Skewness
    • Skewness is a term used to define a distribution’s symmetry. We can immediately determine from density curves if a graph is left, right, or has no skew:
  2. Locate the mean and the median
    • We can rapidly determine whether the mean or median is greater in a particular distribution based on the skewness of a density curve.
    • The mean is less than the median when a density curve is left-skewed
    • The mean is bigger than the median when a density curve is right-skewed.
    • The mean and median are equal when a density curve has no skew.
  3. A number of peaks
    • We can easily determine how many “peaks” there are in a particular distribution by using density curves. If there is only one peak we can categorize those distributions as unimodal.
    • We refer to distributions that contain two peaks as bimodal distributions. Additionally, multimodal distributions with two or more peaks are occasionally possible.

Examples of density curves in real-world applications

  1. Stock market
    • The majority of us have heard of the peaks and valleys in share prices on the stock market. We hear in the news about rising and decreasing share prices. A bell-shaped curve is frequently formed by these variations in the log values of Forex rates, price indexes, and stock prices. The volatility of stock returns is a common name for the standard deviation.
    • More than 99 percent of returns are anticipated to fall within the variance of the mean value if returns are normally distributed. Analysts and investors can draw statistical conclusions about the projected return and risk of stocks thanks to the bell-shaped normal distribution’s features.
  2. Birth weight
    • A newborn’s usual birth weight falls between 2.5 and 3.5 kg. The majority of infants are born at a normal weight; just a small fraction are heavier or lighter than average. Hence, birth weight likewise follows the density curve.
  3. Students’ performance report
    • Schools nowadays promote their performances on social media and television. They persuade parents to enroll their child in that school by displaying the school’s average test score. The average academic achievement of all students is determined by the school administration, and it typically follows the density curve. Average intelligent students outnumber most other students in terms of numbers.

How to find the height of a density curve?

The height of the curve is the value of the peak on the y-axis.

We draw a perpendicular line from the x-axis in such a way it meets the peak of the curve, and the y-value of the point is the height of the density curve.

Conclusion

In this article, we learned about density curves and how they are used in statistics. The density curves are the most important visual representation of probability distribution and can be used to identify the mean, median, and ranges of probabilities of the events occurring. They are used in a lot of day-to-day life analyses and prove to be handy to understand distributions.

Sample examples

Example 1: There is a density curve of how long it takes a student to solve a problem and find the height of the curve.

Solution 1:

The height of the curve is the value of the peak of the curve on the y-axis.

Therefore, from the given graph the height is 0.5

Example 2: There is a given dataset of 10 different heights of students (to the nearest feet). Create a histogram and the normal density curve for the given data.

Data: 4,5,5,5,5,6,4,3,4,3

Solution 2:

Constructing a histogram with their relative frequencies we get

By finding the mean and the standard deviation we can plot the normal density curve.

Mean=\frac{{4 + 5 + 5 + 5 + 5 + 6 + 4 + 3 + 4 + 3}}{{10}} = 4.4

Standard deviation=

    \[\begin{array}{l}{s^2}\begin{array}{*{20}{l}}{\; = \;}&{\frac{{\sum {{{({x_i} - \mathop x\limits^\_ )}^{}}} }}{{N - 1}} = \frac{{{{\left( {4{\rm{ }} - {\rm{ }}4.4} \right)}^2}\; + {\rm{ }}...{\rm{ }} + {\rm{ }}{{\left( {3{\rm{ }} - {\rm{ }}4.4} \right)}^2}}}{{10{\rm{ }} - {\rm{ }}1}} = \frac{{8.4}}{9} = 0.93333}\end{array}\\s = \sqrt {0.93333}  = 0.966091\\\end{array}\]

Plotting the graph we get the normal density curve to be as follows.

Example 3: Find the probability of choosing a value greater than z =  - 0.528orP(z >  - 0.528).

Solution 3:

First, we draw the curve to see where the value lies, we round the score up-to

-0.53

Now we use the Z-score table to look up the value.

Now using the graph, the area on the right is 0.5 and the rest of the shaded region is 0.20194

Therefore, the probability is 0.5 + 0.20194 = 0.70194.

Example 4: Find P( - 1.3 < z < 2.2)

Solution 4:

From the graph with a curve whose area is one, we draw both the values’ lines in the curve.

We get the area of the right half of the score to be 0.4861

And the left half to be 0.4032

Therefore, the probability will be 0.4861 + 0.4032

Example 5: Find P(z < 2.3).

Solution 5:

We use the graph to find the mean which is 0.5and calculate the area around it for the score of 2.3

Since we want to calculate the probability that it is lesser than 2.3 we have to take the area to the left of the z-score which is 0.9893 from the z-score table.

Frequently asked questions (FAQs)

What is the normal distribution?

An example of a continuous probability distribution is the normal distribution, in which the majority of data points cluster in the middle of the range while the remaining ones taper off symmetrically toward either extreme. The distribution’s mean is another name for the center of the range.

What is a probability distribution?

A statistical function called probability distribution explains all the potential values and probabilities for a random variable within a certain range.

What are bimodal distributions?

Two modes comprise a bimodal distribution. In other words, the results of two distinct processes are combined into a single piece of data. The distribution sometimes goes by the name “double-peaked.”

What does the skewness of a curve mean?

When data points on a bell curve are not evenly distributed to the left and right sides of the median, this is known as skewness. The bell curve is referred described as being skewed if it is tilted to the left or right.

What are quartiles?

Three values are called quartiles to divide sorted data into four equal portions with the same number of observations in each.

References

Silverman, B. W. (2018). Density estimation for statistics and data analysis. Routledge.

Gallón, S., Loubes, J. M., & Maza, E. (2013). Statistical properties of the quantile normalization method for density curve alignment. Mathematical biosciences, 242(2), 129-142.

Written by

Prerit Jain

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