Linear interpolation in statistics is a technique for estimating values within two known data points. It's used to calculate key measures such as the median, first quartile (Q1), and third quartile (Q3) in a dataset. By applying the formula y = y1 + ((x-x1)(y2-y1))/(x2-x1), statisticians can predict values that fall within class intervals of grouped data. This method assumes a linear relationship between points and is crucial for interpreting frequency distributions and cumulative frequencies.
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Linear interpolation is a statistical method used to estimate values within two known data points
Frequency distribution with class intervals
Linear interpolation is particularly useful for interpreting grouped data in a frequency distribution with class intervals
Median, quartiles, and percentiles
Linear interpolation can be used to predict values such as the median, quartiles, and percentiles in grouped data
The formula for linear interpolation is y = y1 + ((x-x1)(y2-y1))/(x2-x1), where x1 and y1 are the coordinates of the first data point, x2 and y2 are the coordinates of the second data point, x is the value at which we want to interpolate, and y is the estimated value
To calculate the median using linear interpolation, one must first determine the cumulative frequency for each class interval and plot it against the upper class boundaries
By finding the class interval that includes the median position and using the linear interpolation formula, one can estimate the median value within that interval
The median corresponds to the (n+1)/2th value, where n is the total number of observations
The first quartile, or Q1, is the 25th percentile of a dataset and can be estimated using linear interpolation by locating its position at the (n+1)/4th value
The third quartile, or Q3, is the 75th percentile of a dataset and can be estimated using linear interpolation by locating its position at the 3(n+1)/4th value
To accurately estimate the quartiles using linear interpolation, it is essential to identify the class interval that contains the respective quartile position
Linear interpolation is not limited to calculating the median and quartiles, but can also be used for forecasting and predicting data trends
When dealing with grouped data, constructing a cumulative frequency graph is crucial for the application of linear interpolation
The positions of the median, first quartile, and third quartile are based on the total number of observations and are located at the (n+1)/2th, (n+1)/4th, and 3(n+1)/4th positions, respectively
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Linear interpolation assumption
Assumes linear relationship between two known data points for estimation.
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Linear interpolation application
Used for estimating medians, quartiles, percentiles in grouped data.
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Linear interpolation formula variables
x1, y1: coordinates of first data point; x2, y2: coordinates of second data point; x: interpolation value; y: estimated value.
