Descriptive statistics are widely used in qualitative research. When establishing if a population is normally distributed explain when the statistics of sample size, mean, mode, median, skewness and kurtosis will guide the researcher to make conclusions of a normal distribution.
When determining if a population is normally distributed, several statistics can guide researchers in making conclusions. Let's look at each of these statistics and how they play a role in identifying normal distribution:
1. Sample Size: The larger the sample size, the more reliable the estimate of the population distribution. Large sample sizes tend to provide more accurate data, which helps in properly assessing the normality of the population.
2. Mean: The mean represents the arithmetic average of a set of data. In a normally distributed population, the mean, median, and mode are all equal. So, if the mean closely resembles the median and mode, it suggests that the data is likely normally distributed. However, some caution is necessary since normality is not solely determined by the mean.
3. Mode: The mode refers to the value(s) that occur most frequently in a dataset. In a normal distribution, there is only one mode, and its value will be equal to the mean and median. If the data has a single mode close to the mean, it indicates a likelihood of normality, but it is not sufficient evidence on its own.
4. Median: The median is the middle value in a dataset when the values are arranged in ascending or descending order. In a normal distribution, the median also equals the mean and mode. If the data's median is close to the mean, it further suggests normality, but additional analysis is still required.
5. Skewness: Skewness measures the asymmetry of a distribution. In a normal distribution, the skewness is close to zero (i.e., there is no significant skew). If the skewness value is around zero or falls within a specific range (-1 to +1), it indicates that the data approximates a normal distribution. However, it's important to note that skewness alone may not provide a conclusive answer about the normality.
6. Kurtosis: Kurtosis measures the degree of "peakedness" or the presence of outliers in a distribution. In a normal distribution, the kurtosis is around 0, indicating neither significant flatness (negative kurtosis) nor excessive peakedness (positive kurtosis). So, if the kurtosis value is close to 0, it supports the assumption of normality. However, like skewness, it is not a definitive measure on its own.
To determine normal distribution, it is important to consider all these statistics collectively while also considering the research context and purpose. While these statistics provide some guidance, conducting formal statistical tests like the Shapiro-Wilk or Kolmogorov-Smirnov tests can provide more accurate conclusions about normality.
When establishing if a population is normally distributed, several descriptive statistics can guide the researcher in making conclusions about the distribution. Let's discuss each of these statistics and how they can help in determining normality:
1. Sample size: The larger the sample size, the better representation it provides of the population. With a large enough sample size (usually considered to be above 30), the sample distribution tends to approach a normal distribution due to the Central Limit Theorem. Therefore, a larger sample size increases the likelihood of a normal distribution.
2. Mean: In a normal distribution, the mean, median, and mode are all equal. If the mean of the sample data is approximately equal to the median and mode, it suggests a potential normal distribution. However, it is important to note that a normal distribution is not solely determined by the mean alone.
3. Mode: A normal distribution usually has a single, prominent peak, which corresponds to the mode. If the sample data has a single mode and is symmetrically distributed around that mode, it indicates a potential normal distribution. However, it's important to consider that some distributions other than normal can also have a single mode.
4. Median: The median is the value that divides the data into two equal halves. In a perfectly normal distribution, the median will be equal to the mean and mode. If the data has a symmetric distribution around the median, it suggests a potential normal distribution.
5. Skewness: Skewness measures the asymmetry of a distribution. In a normally distributed population, the skewness is zero or close to zero. Positive skewness indicates a longer right tail, while negative skewness means a longer left tail. If the sample data has skewness close to zero, it suggests a potential normal distribution. However, it is important to consider the sample size because small sample sizes can lead to skewed distributions even if the population is normally distributed.
6. Kurtosis: Kurtosis measures the degree of heaviness of the tails of a distribution compared to a normal distribution. A normal distribution has a kurtosis of zero. If the sample data has kurtosis close to zero, it suggests potential normality. Positive kurtosis indicates more extreme outliers and heavier tails, while negative kurtosis indicates lighter tails. Again, sample size should be considered when interpreting kurtosis.
Keep in mind that these statistics are not definitive measures of normality, but rather indicators that guide researchers in making conclusions about the distribution. It is often beneficial to use multiple statistical tests and graphical methods, such as histograms and probability plots, to assess the normality of a population.