One sandwich is chosen at random (this is a random sample of size one). The weights of the sandwiches are distributed normally with a mean of 150 grams and a standard deviation of 25 grams.
Observation: Click here for additional characteristics of the normal distribution function (using calculus), as well as proofs of Properties 1 and 2.Įxample 2: A charity group prepares sandwiches for the poor. Property 2: If x 1 and x 2 are independent random variables, and x 1 has normal distribution N( μ 1 ,σ 1) and x 2 has normal distribution N( μ 2, σ 2) then x 1 + x 2 has normal distribution N( μ 1+ μ 2, σ) where Property 1: If x has normal distribution N( μ,σ) then the linear transform y = ax + b, where a and b are constants, has normal distribution N( aμ+b, aσ). This will be important when considering the critical value for α =. It also turns out that 95% of the area under the curve is in the interval -1.96 < x < 1.96. The area under the curve for x > μ + 2 σ is 2.28% and the area under the curve for x > μ + 3 σ is 0.13%. the points where the curve changes from concave up to concave down or from concave down to concave up) at x = μ ± σ.Īs can be seen from Figure 2, the area under the curve in the interval μ – σ μ + σ is 15.87%, i.e. The function is symmetric about the mean with inflection points (i.e. Observation: Key properties of the normal distribution are as follows: Since the curve reaches its highest point at 100, it follows that the mode is also 100. Since the normal curve is symmetric about the mean, it follows that the median is also 100. Observation: As can be seen from Figure 1, the area under the curve to the right of 100 is equal to the area under the curve to the left of 100 this makes 100 the mean. As you can see from the figure, the curve has the characteristic bell shape of the normal distribution. You can draw the chart of the probability density curve, as shown on the right side of Figure 1, by highlighting the range B4:B104, selecting Insert > Charts|Line and then making the formatting changes as described in Line Charts. You next insert the formula =NORM.DIST(A4,100,16,FALSE) in cell B4, highlight the range B4:B104 and press Ctrl-D.įigure 1 – Probability density function for IQ This is done by first inserting the value 50 in cell A4, the formula =A4+1 in cell A5, and then highlighting the range A5:A104 and pressing the key sequence Ctrl-D. To create the graph, we first insert the values x = 50, 51, …, 150 in range A4:A104 of Figure 1 and then add the corresponding f( x) values in column B. This distribution is known to be the normal distribution N(100, 256) where 256 is the variance (and so the standard deviation is 16). For earlier versions of Excel, the following functions are used: NORMDIST, which is equivalent to NORM.DIST, and NORMINV, which is equivalent to NORM.INV.Įxample 1: Create a graph of the distribution of IQ scores using the Stanford-Binet scale. These functions are not available for versions of Excel prior to Excel 2010. NORM.INV( p, μ, σ) = the value x such that NORM.DIST( x, μ, σ, TRUE) = p i.e. NORM.DIST( x, μ, σ, cum) = the probability density function value f( x) for the normal distribution N( μ, σ 2 ) when cum = FALSE and the corresponding cumulative distribution function F( x) when cum = TRUE We will also use the notation x ∼ N( µ,σ) or x ∼ N( µ,σ 2) to mean that x is a random variable with mean µ and standard deviation σ (or variance σ 2).Īs we shall see throughout the website, the normal distribution occurs frequently and is very useful in statistics.Įxcel Functions: Excel provides the following functions regarding the normal distribution: We use either the abbreviation N( µ,σ) or N( µ,σ 2) to refer to a normal distribution with mean µ and standard deviation σ (or variance σ 2). It turns out that µ is the mean of the normal distribution and σ is the standard deviation. The normal distribution is completely determined by the parameters µ and σ. Here is the constant e = 2.7183…, and is the constant π = 3.1415… which are described in Built-in Excel Functions.
Definition 1: The probability density function (pdf) of the normal distribution is defined as:
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