Question

Suppose a state government wants to analyze the number of children in families for improving their immunization program. They analyze a group of 200 families and report their findings in the form of a frequency distribution shown in Table \(3.16\) 1\. Draw the bar chart for the following data and calculate the total number of children. 2\. Calculate mean, mode, and median of the data. 3\. Calculate coefficient of kurtosis and coefficient of skewness in the above data. $$ \begin{array}{l} \text { Table 3.16 Frequency table for Problem } 3.8\\\ \begin{array}{l|r|r|r|r|r|l|l|l} \hline \text { No. of children } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text { No. of families } & 18 & 30 & 72 & 43 & 25 & 8 & 3 & 1 \\ \hline \end{array} \end{array} $$

Short answer

After carrying out the data analysis, the total number of children, mean, median, and mode were calculated. Further, the skewness and kurtosis coefficients were computed indicating the asymmetry and peakedness of the data respectively.

Step-by-step solution

Drawing the Bar Chart and Calculating Total Children

Construct a bar chart where the x-axis represents the number of children while the y-axis represents the number of families. As per the data, the height of the bar at 0 will be 18, at 1 will be 30 and so on up to 7. \n For total number of children, multiply the number of children by the number of families and sum all these products. So, it'll be \(0*18 + 1*30 + 2*72 + 3*43 + 4*25 + 5*8 + 6*3 + 7*1\)

Calculating Mean, Mode, and Median

The mean is calculated by dividing the total number of children by the total number of families. The mode is the number of children most families have. Looking at the number of families, most families have 2 children, so the mode is 2. The median is the central data point of the sorted list, which will be the average of 100th and 101st families when sorted. Arrange the dataset in ascending order and identify these points.

Calculating Coefficient of Kurtosis and Coefficient of Skewness

Once the mean and standard deviation are known, compute the skewness and kurtosis coefficients. Skewness is calculated as \[\frac{\sum_{i=1}^{n}(x_{i}-\text{{mean}})^3}{n * s^3}\]where \(x_{i}\) is each value from the dataset, n is number of values, s is standard deviation.\n Kurtosis is calculated as \[\frac{\sum_{i=1}^{n}(x_{i}-\text{{mean}})^4}{n * s^4}\] -3\n\n where \(x_{i}\) is each value from the dataset, n is number of values, s is standard deviation. Coefficient of skewness and kurtosis need to be calculated by plugging the given dataset into the formulas.

Key concepts

Frequency Distribution

A frequency distribution is a summary of data showing the number of occurrences of each value or group of values in a dataset. It is essential for analyzing the distribution patterns that help in understanding the shape and spread of data. This type of distribution can be presented in table form, as in the given exercise, or graphically, such as a bar chart or histogram.

When you create a bar chart from a frequency distribution table, each bar represents the frequency, or number of times, a data point occurs, which helps in visualizing the data easily. Calculating the total number of children across all families involves multiplying the number of children (the value) by the frequency (the number of families) for each category and summing these products. This total will provide insight into the combined child count for the entire group surveyed.

Mean, Mode, Median Calculation

The mean, mode, and median are measures of central tendency that provide information about the distribution of data in a dataset. The mean is calculated by dividing the sum of all data values by the number of values. It represents the average value and is affected by extreme values or outliers.

The mode, on the other hand, is the most frequently occurring value in the dataset and can be located by identifying the value with the highest frequency. It is possible to have more than one mode in a dataset, or none at all if all values occur only once.

The median is the middle point of the data set when the values are arranged in order. If the dataset has an odd number of observations, the median is the value at the exact middle. If the dataset is even, as in the exercise, it is the average of the two middle numbers. To calculate the median from the frequency distribution, one must pinpoint the middlemost frequency and identify the corresponding data value.

Coefficient of Skewness and Kurtosis

Skewness is a measure of the symmetry, or lack thereof, in the dataset. A skewness of zero suggests a symmetrical distribution, while a positive or negative skew indicates data that are spread out more to the right or left, respectively. The coefficient of skewness is calculated using the third moment of the distribution and shows how much the distribution deviates from a normal distribution in terms of asymmetry.

Kurtosis, on the other hand, measures the ‘tailedness’ of the distribution - that is, the presence of outliers. A high kurtosis indicates more outliers and a distribution with heavy tails, while a low kurtosis indicates a flatter distribution. It is calculated using the fourth moment of the distribution. Like skewness, deviations from the normal distribution (with a kurtosis value of 3) are what is being measured, so often '3' is subtracted from the result to center the value around zero for a normal distribution.

Calculating these coefficients requires the mean and standard deviation, which are derived from the dataset. Once calculated, these coefficients help describe the shape of the data distribution beyond just the center and spread.