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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.
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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 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.
The shape of density curves
Steps for creating a density curve from data
We can interpret the density curves in different ways:
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.
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.
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=
Standard deviation=
Plotting the graph we get the normal density curve to be as follows.
Example 3: Find the probability of choosing a value greater than or
.
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 and the rest of the shaded region is
Therefore, the probability is .
Example 4: Find
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
Example 5: Find .
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.
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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.
A statistical function called probability distribution explains all the potential values and probabilities for a random variable within a certain range.
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.”
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.
Three values are called quartiles to divide sorted data into four equal portions with the same number of observations in each.
Silverman, B. W. (2018). Density estimation for statistics and data analysis. Routledge.