When analyzing data, the choice of summary statistics and methods may vary depending on the distribution of the data. Here's a general guideline:

For Normally Distributed Data:

Mean: The mean is a commonly used measure of central tendency for normally distributed data. It represents the average value of the data.

Standard Error: The standard error is used to measure the precision of the estimate of the mean. It is calculated as the standard deviation divided by the square root of the sample size.

For Not Normally Distributed Data:

Median: The median is an alternative measure of central tendency that is not affected by extreme values or skewed distributions. It represents the middle value of the data.

Percentiles: Percentiles are used to describe how values are distributed across the data. They divide the data into equal parts, such as quartiles (25th, 50th, and 75th percentiles) or deciles (10th, 20th, etc. percentiles).

It's important to note that these guidelines are not strict rules, but rather general practices. The choice of summary statistics depends on the specific context and the nature of the data. Different analyses and research fields may have their own conventions and requirements. It is also advisable to consider the shape of the distribution and conduct appropriate tests or consult with domain experts if needed.

Normal distribution's characteristic function is defined by just two moments:

mean and the variance (or standard deviation).

Therefore, for normal distribution the standard deviation is especially important, it's 50% of its definition in a way.

For other distributions the standard deviation is in some ways less important because they have other moments. However, for many distributions used in practice the first few moments are the largest, so they are the most important ones to know.

Now, intuitively, the mean tell you where the center of your distribution is, while the standard deviation tell you how close to this center your data is.

Since the standard deviation is in the units of the variable it's also used to scale other moments to obtain measures such as kurtosis.

Kurtosis is a dimensionless metric which tells you how fat are the tails of your distribution compared to normal

Thanks for replly as i get from your valuable answers for both normaly and not normaly distributed data we can use mean to discuss our result

Your goal is to accurately describe the data. Most researchers pick measures of central tendency. Consider all: the arithmetic mean (other means might be more appropriate), median, mode, and SD. Plotting data helps. For example, outliers can make the arithmetic mean not an accurate descriptor.