1. **Step in computing variance and standard deviation using Excel for ungrouped data:** - Suppose your data is in cells A1 to A10. - To find the mean, use the formula: $$\text{Mean} = \text{AVERAGE}(A1:A10)$$ - To find the variance, use: $$\text{Variance} = \text{VAR.S}(A1:A10)$$ (for sample variance) - To find the standard deviation, use: $$\text{Standard Deviation} = \text{STDEV.S}(A1:A10)$$ 2. **Z test (Computation of one sample mean and two sample mean):** - For one sample mean, the Z test statistic is: $$Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$$ where $\bar{x}$ is sample mean, $\mu$ is population mean, $\sigma$ is population standard deviation, and $n$ is sample size. - For two sample means, the Z test statistic is: $$Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$ where subscripts 1 and 2 refer to the two samples. 3. **Variance:** - Variance measures the average squared deviation from the mean. - Formula for sample variance: $$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$ 4. **Standard Deviation:** - Standard deviation is the square root of variance: $$s = \sqrt{s^2}$$ - It measures the average distance of data points from the mean. 5. **Mean Absolute Deviation (MAD):** - MAD is the average of absolute deviations from the mean: $$\text{MAD} = \frac{1}{n} \sum_{i=1}^n |x_i - \bar{x}|$$ These formulas and Excel functions help compute these statistics efficiently.