Standard Deviation :

• $\sigma$$\sigma$ represents the standard deviation.

• $N$$N$ = the number of data points.

• $x_i$$x_i$ is the $i^{th}$$i^{th}$ data point in the dataset

• $\mu$$\mu$ is the mean ( average ) of the dataset

  • given by $\mu =\frac{1}{N}\sum _{i=1}^N{x}_i$$\mu = \frac{1}{N} \sum_{i=1}^{N} x_i$

• The sum $\sum _{i=1}^N(x_i-\mu {)}^2$$\sum_{i=1}^{N} (x_i - \mu)^2$ calculates the squared deviation of each data point from the mean , and then sums these squared deviations.

• The square root $\sqrt{\cdot }$$\sqrt{\cdot}$ is taken to bring the units back to the original units of the data , since squaring the deviations had squared the units

Standard Normal Distribution :

• This refers to a specific type of probability distribution.

• It is a normal distribution (a type of bell curve) that has a mean of $0$$0$ and a standard deviation of $1$$1$.

• The graph of a standard normal distribution is a bell-shaped curve where :

  • the x-axis represents the z-scores

  • the y-axis represents the probability density

Cumulative Distribution Function (CDF) :

• The CDF of any probability distribution, including the standard normal distribution,

  • is a function that gives the probability that a random variable is less than or equal to a certain value.

• In the case of the standard normal distribution, the CDF is denoted as $\mathrm{\Phi }(z)$$\Phi(z)$ , where $z$$z$ is a z-score. The function $\mathrm{\Phi }(z)$$\Phi(z)$ gives the probability that a standard normal random variable is less than or equal to $z$$z$

Confidence Level : The confidence level is the probability that the true value of a parameter lies within the calculated confidence interval.

• Common choices for the confidence level are 90%, 95%, and 99%.

• For instance, if you set a confidence level of 95%, you are stating that if the same population were sampled multiple times, approximately 95% of the resulting confidence intervals would contain the true population parameter.

Confidence Interval : The confidence interval is the range of values, derived from the statistical analysis, that is likely to contain the true parameter with a specified level of confidence.

• It provides a range estimate for unknown population parameters.

• The width of the confidence interval gives us some idea about how uncertain we are about the unknown parameter.

• A wider interval might indicate that we need more data to get a precise estimate of the parameter, while a narrow interval might suggest our estimate is quite accurate.

The CDF of the standard normal distribution, often denoted by $\mathrm{\Phi }(z)$$\Phi(z)$ , is given by :

This returns :

• the area under the curve from negative infinity to $z$$z$

• the probability that a random variable from a standard normal distribution is less than or equal to a certain value $z$$z$

Z-Score Example : To find the z-score such that $95\mathrm{\%}$$95\%$ of the data falls within $+z$$+z$ and $-z$$-z$ , we need to solve the following for $z$$z$ :

Because the standard normal distribution is symmetric around 0 :

Therefore, the equation becomes :

Solving for $\mathrm{\Phi }(z)$$\Phi(z)$ :

So we're looking for the z-score where the cumulative distribution function $\mathrm{\Phi }(z)$$\Phi(z)$ equals $0.975$$0.975$

This is the step that involves the inverse cumulative distribution function , or the percent-point function.

• This function is generally denoted ${\Phi }^{-1}$$Φ^{-1}$ , so we solve :

The problem is , there's no simple closed-form solution for the inverse of the cumulative distribution function $\mathrm{\Phi }(z)$$\Phi(z)$ for the standard normal distribution.

The standard normal cumulative distribution function $\mathrm{\Phi }(z)$$\Phi(z)$ involves an integral that cannot be expressed in terms of elementary functions

In the computation of ${\mathrm{\Phi }}^{-1}(0.975)$$\Phi^{-1}(0.975)$ , we will make use of the relationship with the inverse error function $I(x)$$I(x)$.

• This is because the standard normal distribution is symmetric around $0$$0$ ,

  • and the error function , commonly used in statistics , is a similar function ,

    • but defined over $[0,1]$$[0, 1]$ rather than $[-0.5,0.5]$$[-0.5, 0.5]$

• For a probability $p$$p$ of $0.975$$0.975$ , the input to the inverse error function $I(x)$$I(x)$ would be $0.975-0.5=0.475$$0.975 - 0.5 = 0.475$

Excel or some calculator software will just do a series approximation to compute the value of the inverse CDF

Simple Series approximation :

• where $I$$I$ is the inverse error function , usually denoted as ${\text{erf}}^{-1}(x)$$\text{erf}^{-1}(x)$ , and is given by :

• This is a power series that is summed over all integer values of $n$$n$ from $0$$0$ to infinity

• This power series provides a mathematical approximation for the inverse error function , which can be used to compute the inverse of the cumulative distribution function of the standard normal distribution.

• However , this series converges very slowly and is not practically used for numerical computation.

• Instead, other faster-converging numerical approximations are used in software and tables.

• The implementation details of these faster methods are usually more complex and are beyond the scope of a basic understanding of statistical theory.

The confidence level corresponds to a Z-score, which is a constant value for any given confidence level

• it technically is the number of standard deviations a data point is away from the mean

Here are the Z-scores for the most common confidence intervals :

• 90% - Z score = 1.645

• 95% - Z score = 1.96

• 99% - Z score = 2.576

To calculate an appropriate sample size :

Example of a $95\mathrm{\%}$$95\%$ confidence level , $0.5$$0.5$ standard deviation , and a confidence interval of $\pm 5\mathrm{\%}$$\pm 5\%$ :

• 95% confidence level :

  • if you conducted the same experiment 100 times, you'd expect the true value to fall within your calculated range 95 times

• with a confidence interval of $\pm 5\mathrm{\%}$$\pm 5\%$ ,

  • if the mean is estimated to be 16 , the range would be from 15.2 to 16.8 ,

    • suggesting that we are 95% confident the true mean falls within this range