Statistics Chapter 3 Exercise Answers
Brook Mithani
NCERT Solutions for Class 10 Maths Chapter 14 Statistics are provided here, which can be downloaded for free, in PDF format. The solutions are prepared by our subject experts who have mastery in Maths. All the solved questions of Statistics are with respect to the latest updates on the CBSE syllabus and guidelines, to help students solve each exercise question and effectively prepare for the CBSE exam.
Using these solutions as a reference tool will be helpful for the students to score good marks. Students can also get the exercise-wise Solutions for Class 10 Maths in all chapters and practise solving the problems.
1. A survey was conducted by a group of students as a part of their environment awareness program, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.
4. Thirty women were examined in a hospital by a doctor, and the number of heartbeats per minute were recorded and summarised as follows. Find the mean heartbeats per minute for these women, choosing a suitable method.
5. In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.
3. The life insurance agent found the following data for the distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to the persons whose age is 18 years onwards but less than the 60 years.
6. 100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:
Locate the point 17.5 on the y-axis and draw a line parallel to the x-axis cutting the curve at a point. From the point, draw a perpendicular line to the x-axis. The intersection point perpendicular to x-axis is the median of the given data. Now, to find the median by making a table.
Class 10 Maths Chapter 14, Statistics, is one of the most important chapters present in the textbook. The weightage of this chapter in the CBSE exam is around 11 to 12 marks. On average, there will be 3 questions which could be asked from this chapter and marks will be distributed in a manner of 3+4+4 (it could vary as per question).
The world is highly data-oriented, in fact, each and every field has a group of data, which represents the relevant information. Statistics is the branch of mathematics which deals with the representation of data in a meaningful way.
You will face many real-life scenarios where the fundamentals of statistics are used to represent a set of data in tabular form, in graphs or in pie charts. There are a number of methods you will learn from this chapter such as, step deviation methods, finding mode and median of grouped data, converting frequency distribution and the relation between mode, mean and median methods, etc. 10th Class NCERT solutions are the best study materials to prepare for the CBSE exam.
Think Stats is an introduction to Statistics and Data Science for Python programmers.If you have basic skills in Python, you can use them to learn concepts in probability and statistics and practical skills for working with data.
The easiest way to work with this code it to run it on Colab, which is afree service that runs Jupyter notebooks in a web browser.For every chapter, I provide two notebooks: one contains the code from thechapter and the exercises; the other also contains the solutions.
Statistics has become the universal language of the sciences, and data analysis can lead to powerful results. As scientists, researchers, and managers working in the natural resources sector, we all rely on statistical analysis to help us answer the questions that arise in the populations we manage. For example:
These are typical questions that require statistical analysis for the answers. In order to answer these questions, a good random sample must be collected from the population of interests. We then use descriptive statistics to organize and summarize our sample data. The next step is inferential statistics, which allows us to use our sample statistics and extend the results to the population, while measuring the reliability of the result. But before we begin exploring different types of statistical methods, a brief review of descriptive statistics is needed.
Populations are characterized by descriptive measures called parameters. Inferences about parameters are based on sample statistics. For example, the population mean () is estimated by the sample mean (x̄). The population variance (σ2) is estimated by the sample variance (s2).
Variables are divided into two major groups: qualitative and quantitative. Qualitative variables have values that are attributes or categories. Mathematical operations cannot be applied to qualitative variables. Examples of qualitative variables are gender, race, and petal color. Quantitative variables have values that are typically numeric, such as measurements. Mathematical operations can be applied to these data. Examples of quantitative variables are age, height, and length.
Descriptive measures of populations are called parameters and are typically written using Greek letters. The population mean is μ (mu). The population variance is σ2 (sigma squared) and population standard deviation is σ (sigma).
Descriptive measures of samples are called statistics and are typically written using Roman letters. The sample mean is (x-bar). The sample variance is s2 and the sample standard deviation is s. Sample statistics are used to estimate unknown population parameters.
The population mean is represented by the Greek letter μ (mu). The sample mean is represented by x̄(x-bar). The sample mean is usually the best, unbiased estimate of the population mean. However, the mean is influenced by extreme values (outliers) and may not be the best measure of center with strongly skewed data. The following equations compute the population mean and sample mean.
The median of a variable is the middle value of the data set when the data are sorted in order from least to greatest. It splits the data into two equal halves with 50% of the data below the median and 50% above the median. The median is resistant to the influence of outliers, and may be a better measure of center with strongly skewed data.
The mode is the most frequently occurring value and is commonly used with qualitative data as the values are categorical. Categorical data cannot be added, subtracted, multiplied or divided, so the mean and median cannot be computed. The mode is less commonly used with quantitative data as a measure of center. Sometimes each value occurs only once and the mode will not be meaningful.
Understanding the relationship between the mean and median is important. It gives us insight into the distribution of the variable. For example, if the distribution is skewed right (positively skewed), the mean will increase to account for the few larger observations that pull the distribution to the right. The median will be less affected by these extreme large values, so in this situation, the mean will be larger than the median. In a symmetric distribution, the mean, median, and mode will all be similar in value. If the distribution is skewed left (negatively skewed), the mean will decrease to account for the few smaller observations that pull the distribution to the left. Again, the median will be less affected by these extreme small observations, and in this situation, the mean will be less than the median.
Measures of center look at the average or middle values of a data set. Measures of dispersion look at the spread or variation of the data. Variation refers to the amount that the values vary among themselves. Values in a data set that are relatively close to each other have lower measures of variation. Values that are spread farther apart have higher measures of variation.
Examine the two histograms below. Both groups have the same mean weight, but the values of Group A are more spread out compared to the values in Group B. Both groups have an average weight of 267 lb. but the weights of Group A are more variable.
The variance uses the difference between each value and its arithmetic mean. The differences are squared to deal with positive and negative differences. The sample variance (s2) is an unbiased estimator of the population variance (σ2), with n-1 degrees of freedom.
The standard deviation is the square root of the variance (both population and sample). While the sample variance is the positive, unbiased estimator for the population variance, the units for the variance are squared. The standard deviation is a common method for numerically describing the distribution of a variable. The population standard deviation is σ (sigma) and sample standard deviation is s.
We want to use this sample mean to estimate the true but unknown population mean. But our sample of 100 trees is just one of many possible samples (of the same size) that could have been randomly selected. Imagine if we take a series of different random samples from the same population and all the same size:
The sample mean (x̄) is a random variable with its own probability distribution called the sampling distribution of the sample mean. The distribution of the sample mean will have a mean equal to and a standard deviation equal to .
To compare standard deviations between different populations or samples is difficult because the standard deviation depends on units of measure. The coefficient of variation expresses the standard deviation as a percentage of the sample or population mean. It is a unitless measure.
Variability is described in many different ways. Standard deviation measures point to point variability within a sample, i.e., variation among individual sampling units. Coefficient of variation also measures point to point variability but on a relative basis (relative to the mean), and is not influenced by measurement units. Standard error measures the sample to sample variability, i.e. variation among repeated samples in the sampling process. Typically, we only have one sample and standard error allows us to quantify the uncertainty in our sampling process.
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