Select the T ools distributions. YEAR. N would also be discrete if it represented the number of r ooms in a house. Cambridge Pre-U Mathematics (9794) and . Formula (10) is same as σ = 1 N N 2 ( )2 f x f xi i i i− (11) (g) Another formula for standard deviation : σ x = N h ( ) N 2 2 f y f yi i− (12) where h is the width of class intervals and y i = xi A h − and A is the assumed mean. There are three essential characteristics of descriptive statistics we need to discuss: scales of measurement, measures of central tendency, and measures of v. ) , X discrete. { Mathematical routines analyse the proba-bility of getting certain data, given a par-ticular model. Nowadays, statistics are widely used in data Aug 23, 2021 · Sample: Some data randomly chosen from population. =. Median. The mean, also known as the average, is a measure of central tendency in statistics. 1: Basic Definitions and Concepts is shared under a CC BY-NC-SA 3. Expected values The expected value Of a function H (X) Of a random vari- able X is defined as = an. Jun 14, 2024 · Statistics in Maths. Example: Assume a data set follows a normal distribution with a mean of 50 and a standard deviation of 10. 69. Included in this chapter are the basic ideas and words of probability and statistics. There are four categories we will consider Statistics is a branch of mathematics, which deals with all aspects of data. 55 to 0. STATISTICS FORMULASMEAN: The mean, symbolized by x-bar, equals one divided by the number of samples multiplied by the sum of all data points, sy. The mean add up all the numbers and divide by how many numbers there are. In our research the focus in on descriptive statistics. It is based on literature [1, 6, 3] and in-class material from courses of the statistics department at the Univer-sity of California in Berkeley but also in uenced by other sou. is the symbol used to indicate that values are to be summed (see Sigma Notation) we. Here are some basic probability formulas: Probability of a single event: If an experiment has equally likely outcomes, the probability of a specific event E is given by: P(E) = number of favorable outcomes / total number of possible outcomes. In simpler words, it is a field to collect and summarize data. Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. 13. With the help of statistics, we are able to find various measures of central tendencies and the deviation of data values from the center. Expectation is linear in that the expectation Of a linear combination Of functions is the same linear combination Of expectations. • Step 2: Use this formula to find a set of Scheffe values you will use in the next step. Find the mode. review ed by Dave K och and Emily Ryan (Emory. Mean (Arithmetic Mean) To calculate the arithmetic mean of a set of data we must first add up (sum) all of the data values. The material supports students with Statistics is a branch of science dealing with collecting, organizing, summarizing, analysing and making decisions from data. Enter the values for n, p and x into each cell. descriptive statistics for a sample data: 2, 4, 6, and 8. Average the middle 2 values. This includes: Types of data (categorical and numerical) Type of frequency distributions. You randomly select a sample of 11th graders in your state and collect data on their SAT scores and other characteristics. COUNT function. Formulas and T erms. ? Bayesian: { There are no true model parameters. Here are some examples: Examples from Chemistry: What is thermal equilibrium? statistics to get a feeling for the data. e. Excel Function List We also have a large list of example formulas and video training. Suppose, for example, we want to find the probability of getting 4 heads in 10 tosses. Here, k is the index of summation, 1 is the lower limit, and n is the upper limit. Statistics are the analytical backbone of science. 90; medium, 0. Here are two contrasting definitions of what statistics is, from eminent professors in the field, some 60+ years apart: "Statistics is the branch of scientific method which deals with the data obtained by counting or measuring The formula in statistics: For almost all statistical computations, the basic concepts of mean, median, mode, variance, and standard deviation are the stepping stones. It is denoted by f (x). 2 The Branches of Statistics 1. Jan 10, 2024 · The free cheat sheet covers key concepts and terminology for both descriptive and inferential statistics. by Anders Kallner. Parameters. Calculate this value for each pair. This cheat sheet will make things easy when getting ready for The variance of X, denoted Var(X), is de ned by Var(X) = E[(X )2]. Oct 24, 2023 · The interquartile range or IQR is the difference between the third and first quartiles, which contains the middle 50% of the data. Vertex Form y= a(x h)2 + k or f(x) = a(x h)2 + k In the above example, a confidence level of 95% was selected. 1 Null and Alternative Hypotheses; 9. I hope and wish that the Statistics Formula Sheet and Tables 2020. •• Inferential statistics: statistics used to interpret the meaning of descriptive statistics. mean then squared. Unit could be “Y” for years, “M” for months or “D” for days. Number of Plants. It is based on a more complete list of Excel functions here. Alpha scores are conventionally interpreted as follows: high, 0. 4) Computational Formula: σ= √ ∑X2− :∑X ;2 N R Basic Formula: σ= √SS R Computational Formula: s= √ ∑X2− ∑X ;2 n n − 1 Basic Formula: s= √SS n − 1 D) Variance (Ch. Sep 20, 2023 · Better Education - Statistics discount textbook and endless questions package. 3 Distribution Needed for Hypothesis Testing; 9. 👉🏻The updated probability of the subsequent, or conditional, event is multiplied by the probability of the preceding, or conditional, event to get conditional probability. The importance of this distinction will be discussed later. We can get a fairly good idea about the whole set of An average is a measure of the centre of the data set. Variance is one way to measure the amount a random variable \varies" from its mean over successive trials. larizing statistics among the American public. INTRODUCTION TO STATISTICS Statistics is the branch of mathematics that deals with the techniques for collecting, analyzing, and drawing conclusions from data (Snedecor and Cochran, 1980). Set the significance level. Unlike some other statistical measures of variability, it incorporates all data points in its calculations by contrasting each value to the mean. Step 2: Find the deviation of x i from a, i. In-stead all parameters are treated as random variables with probability distributions. Introduction. Function. (The subtlety here is that in c and u charts, a single unit may have more than one defect, while with p and np charts, an item is either defective or not. Further Mathematics (9795) For use from 2017 in all papers for the above syllabuses. MST-002 DESCRIPTIVE STATISTICS S. This web page lists statistics formulas used in the Stat Trek tutorials. Use the 1-sample analysis to determine whether a population mean is different from a hypothesized value. Statistics for Social Scientists Quantitative social science research: 1 Find a substantive question 2 Construct theory and hypothesis 3 Design an empirical study and collect data 4 Use statistics to analyze data and test hypothesis 5 Report the results No study in the social sciences is perfect Use best available methods and data, but be aware ics describe the data. The central tendency concerns the averages of the values. All data values have the same number of repeats. This resulted in a smaller standard deviation. 1 Statistics is divided into two main areas, which are descriptive and inferential statistics. Design - Planning/Implementing a study. Set Theory The old notion of: is (are) now called: Universal set Ω Sample space Elements of Ω(its individual ’points’) Simple events (complete outcomes) Introduction to Statistics, Data and Statistical Thinking 1. Complement of Event A: For mutually exclusive outcomes: Where not mutually exclusive For Independent outcomes: For dependent outcomes: P ( A _ and _ B ) P ( A )* P ( B A ) Discrete variables: =E(X) Normal Distribution. However, there are two important and basic ideas involved in statistics; they 3 days ago · Mean. They are the mean, the median and the mode and are calculated as follows. The value of z* for a specific confidence level is found using a table in the back of a statistics textbook. It’s best to use a table. In statistical mathematics, the median of a data set of values is the centermost value of the data set when Variance is a measure of variability in statistics. Test Statistics • Test Statistic for Population Proportion 𝑧𝑧= 𝑝𝑝 − 𝑝𝑝 0 𝑝𝑝0(1−𝑝𝑝0) 𝑛𝑛 • Test Statistic for population Mean t = 𝑥𝑥̅− 𝜇𝜇 0 𝑠𝑠 √𝑛𝑛 o with degrees of freedom = df=n-1 Chart for number of defects/nonconformances per sampling unit – c Chart. Statistics in Maths: Statistics constitutes a mathematical discipline focused on gathering, scrutinizing, interpreting, and portraying data. There are 3 main types of descriptive statistics: The distribution concerns the frequency of each value. Since Excel recognizes adjacent rows and columns of data as a dataset, a blank row or column indicates the end of the data set, which can give partial views of the complete data set. 1 The Five Basic Words of Statistics 1. , beginning date, ending date, etc. in@icir. We begin by introducing two general types of statistics: •• Descriptive statistics: statistics that summarize observations. Inferential statistics such as t-tests, chi-square and correlation. FORMULAS Related Links: Area And Perimeter Formula For All Shapes Pdf: Ab Formula: C To F Conversion Formula: Chemistry Formulas Pdf Download: Centigrade To Fahrenheit Formula: Heat Conduction Formula: Tangent Formula: X And Y Intercept Formula: Sample Mean Formula: Z Test Formula Statistics is a branch of mathematics used to summarize, analyze, and interpret a group of numbers or observations. From descriptive statistics we are considering mean, mode, median, quartiles, variance and Introduction to Statistics Basic Statistical Formulas Commonly used Ecological Equations. Guidelines for Data Structure 3 days ago · Basic Statistics Formulas. Statistics is a study of data: describing properties of data (descriptive statistics) and drawing Feb 8, 2024 · It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. Example: End Date October 12, 2015 minus Start Date March 31, 2015 = 195 days Apr 14, 2021 · functions you should know, with links to detailed examples. Surface Area and Volumes Formulas For Class 10. It is calculated by adding up all the values in a dataset and then dividing by the number of values. Find out how to calculate the mean, standard deviation, and z-scores of a normal distribution, and how to compare it with other distributions. A Z test is a form of inferential statistics. The variability or dispersion concerns how spread out the values are. Step 1: Write your hypotheses and plan your research design. . Feb 21, 2024 · It helps us collect, organize, analyze, and interpret data to find patterns, trends, and relationships in the world around us. 4) Definitional Formula: σ2= ∑ :X− Sep 13, 2023 · Basic Statistics Formulas. The goal of research is often to investigate a relationship between variables within a population. Learn the definitions, formulas, and solved examples of these concepts with Testbook, the leading online learning platform for competitive exams. Provides methods for. Helpful tool for studying statistics formulas frequently used statistics formulas and tables chapter class width highest value lowest value (increase to next Statistics Formulas Used on Stat Trek. Population mean = μ = ( Σ X i) / N; Population standard deviation = σ = sqrt [ Σ ( X i - μ ) 2 / N ] , A vs , vs , for example). Class Median is the first class with the value of cumulative frequency equal at least n/2. Step 1: Calculate the central tendency measures to find the mean deviation and let it be ‘a’. , x n – a. PREFACE AboutIntroductory Statistics IntroductoryStatisticsis designed for the one-semester, introduction to statistics course and is geared toward students Jul 9, 2020 · Types of descriptive statistics. Chart for number of defects/nonconformances per sampling unit - u Charts. There are three common ways of describing the centre of a set of numbers. If we have an 102 2x2 Augmented Matrix Examples 103 3x3 Augmented Matrix Example 104 Characteristic Equation and Eigenvalues 105 Eigenvectors 106 2x2 Eigenvalues and Eigenvectors – General Case 107 Calculating Inverses Using Characteristic Equations Chapter 15: Exponents and Logarithms 109 Exponent Formulas 110 Logarithm Formulas 111 e 4 days ago · Add the terms to find the sum. For The probability density function is defined as an integral of the density of the variable density over a given range. AVERAGEA. 1 Bar graphs and frequency tables. As an example: creating tables, sorting, and /or filtering data will not work properly if there are gaps in the data. 18 hours ago · Notice that the formula for sample variance, `s^2`, is identical to the formula for population variance, except that we've swapped out `mu` for `bar x` (since `bar x` is sample mean, whereas `mu` is population mean), and we've changed N to n (since n refers to sample size, whereas N refers to population size). When there is no variability in a sample, all values are the same, and Jun 16, 2024 · Excel has more than 450 functions for various purposes. Press [ENTER]. Median and InterquartileRange – Grouped Data. =YEAR(Date) Returns the year portion of date. For example: Where is the center of the data located? How spread out is the data? How correlated are the data from two variables? The most common descriptive statistics are in the following table, along with their formulas and a short description of what each one measures. The text focuses on diverse applications from a variety of fields and societal contexts, including business, healthcare, sciences, sociology, political science, computing, and several others. t of a study variable. Th eir comments and feedbacks were equally important and useful in making this book an excellent guide into the often-challenging fi elds of mathematics and statistics. 96. Since. In this Statistics cheat sheet, you will find simplified complex statistical concepts, with clear explanations, practical examples, and essential formulas. After putting the value of z*, the population standard deviation, and the . X continuous. g. 4 Sampling Concepts 1. 5 Sample Selection Methods One-Minute Summary Test Yourself E very day, you encounter numerical information that describes or analyzes some aspect of the world you live in. Consistency rating: 5 The explanations are consistent and build from and relate to corresponding sections that are listed in each unit. Class 10 maths statistics mean am = a1 + a2 + a3 + a4 + a5/5 Σno/n. Statistics and probability also play explicit roles in our understanding and modelling of diverse processes in the life sciences. The distribution of cases across the attributes of any variable can be described using a variety of statistics. Among them, the following 111 formulas are the most common and crucial for statistical uses. 4 Rare Events, the Sample, and the Decision and Conclusion For example, if N represents the number of students in a statistics class, N would be a discrete variable. The common formulas from the surface area and volumes chapter in 10 th class include the following: Here, l = length, b = breadth and h = height. Keep in mind that k is condition number, F is the F value found from your ANOVA calculation, and MSE is the Solved Examples on Statistics Formulas. Also, Check: Previous Years IIT JEE Questions on Statistics Quadratic Functions and Formulas Examples of Quadratic Functions x y y= x2 parabolaopeningup x y y= x2 parabolaopeningdown Forms of Quadratic Functions Standard Form y= ax2 + bx+ c or f(x) = ax2 + bx+ c This graph is a parabola that opens up if a>0 or down if a<0 and has a vertex at b 2a;f b 2a . If you leave off the x value and just enter n and p, you will get all the probabilities for each x from 0 to n. Each formula links to a web page that explains how to use the formula. Since there is only one value repeating itself, it is a unimodal list. riability (or spread). Aug 17, 2023 · For example, the variance formula uses "M" for mean, rather than x-bar. Step 1: Construct the cumulative frequency distribution. ized by x-sub-i. 15. √ : G−1 ;∗ ∗ 𝑆 ∗ @1 1 +1 2 A. Example 1: Age of students = {14,15,16,15,17,15,18}. Step 2: Decide the class that contain the median. The value of z* for a confidence level of 95% is 1. Computes the arithmetic average of its argument numbers. For example: Where is Apr 19, 2020 · This reference book on applied statistics for hospitals was designed and developed to provide an elementary understanding of the terms, definitions, and formulae used in computing the hospital’s The fields of economics, business, psychology, education, biology, law, computer science, police science, and early childhood development require at least one course in statistics. Writing statistical hypotheses. I. Cheat Sheet. (b) If you scored 175 on the exam, what would your z-score be? From the formula, z= x ˙ = 175 150 10 = 2:5. The fields of economics, business, psychology, education, biology, law, computer science, police science, and early childhood development require at least one course in statistics. Mar 26, 2023 · This page titled 1. A survey was conducted by a group of students as a part of their environment awareness programmes, in which they collected the following data regarding the number of plants in 30 houses in a locality. Feb 27, 2023 · 1. 89; and low, 0. Descriptive statistics such as the mean, median and mode. 3 Sources of Data 1. To solve statistical problems, there are few formulas of statistics that will be used the most, they are as follows : Mean: To calculate the mean of a given data set, we use the following formula, Mean (\[\bar{x}\]) = \[\frac{\sum x}{N}\] Median: In the case of the median, we have two different formulas. Let us learn the basic statistics formulas along with a few solved examples. Find the mean number of plants per house. example were much closer to the mean than those in the first example. org6th March, 2011 This cheat sheet integrates a variety of topics in probability the-ory and. =DAYS This is a handy formula to calculate the number of days between two dates (so there’s no worries about how many days are in each month of the range). Understanding Formulas for Common Statistics - dummies After data has been collected, the first step in analyzing it is to crunch out some descriptive statistics to get a feeling for the data. STANDARD Sep 4, 2020 · Example: Inferential statistics. If you are new to Excel formulas and how functions are used, see this introduction. 1 day ago · Probability and statistics are two key concepts in mathematics. Step 1: Classify the data into intervals and find the corresponding frequency of each class. What Are the Basic Statistics Formulas? The important statistics formulas are listed in the chart below: \(\begin{array}{|c|c|} function (pdf) f that satisfies f (x) O and f = 1, with P(a < < b) = f (x)dx. For example, here are some news items Oct 23, 2020 · What is a normal distribution and how to use it in statistics? Learn the definition, formulas, examples, and applications of this common data pattern. , x 1 – a, x 2 – a, x 3 – a,. Solved Examples Example 1: First, we have to determine the probability of one possible way the event can occur, and then determine the number of different ways the event can occur. Scribbr offers clear and concise explanations, diagrams, and calculators to help you master this topic. N/2 = 6/2 = 3 so average the 3rd and 4th values. statistics. a single, unidimensional, latent construct in a scale or inven-tory. In statistical mathematics, the mean value of a variable is described as the sum total of all the values of the variable divided by its number of values. Example 2:In a class of 50, 4 students were selected at random and their total marks in the final assessments are recorded, which are: 812, 836, 982, 769. No. That is, P(Event) = (Number of ways event can occur) * P(One occurrence). Scales of measurement are different ways to classify the measureme. Example =YEAR(7/16/2005) would return 2005. A Descriptive Statistics Suppose that a test in statistics course is given to a class at KSU and Statistics. Use a Z test when you need to compare group means. 1. 2 Coefficient of variation It is sometimes useful to describe variability by Business MBA students who studied business mathematics and statistics using the framework laid out in this book. The definition of what is meant by statistics and statistical analysis has changed considerably over the last few decades. Definition 1. Stat 101 Formulas Brian Powers, Summer 2014 Sample Statistics Sample mean ̅= 1 In statistics, we normally use words like “mean”, ”median” and “mode” to understand the data distributed throughout a range. Probability. VARIANCE: Variance, symbolized by s squared, equals 1 divided by the number of samples minus one, multiplied by the sum of each data point subtracted by th. It uses samples to draw conclusions about populations. It aids in the representation of complex data in a simple and understandable manner. These are typically processes where the outcome is influenced by many factors, each with small effect, but with significant total impact. { Random errors in data have no probability distribution, but rather the model param- Jun 10, 2015 · Introductory Statistics 2e provides an engaging, practical, and thorough overview of the core concepts and skills taught in most one-semester statistics courses. 2 Outcomes and the Type I and Type II Errors; 9. The data value with more repeats than any other data value. Decide the Test Statistic (z,t) Find out the critical value. ISBLANK =ISBLANK(value) If a cell is blank, it returns TRUE. ) CL = c. We introduced above the formula Var(X) = E[(X )2]. This notation asks to find the sum of Xk from k=1 to k=n. This is the probability density function and will return you the probability of exactly x successes. AVEDEV. (x) and then divide the result by the number of values (n). Be it grouped or ungrouped, usually, these central tendency measures give a single data point which typically shows the behaviour of the whole data. This function is positive or non-negative at any point of the graph, and the integral, more specifically the definite integral of PDF over the entire space is always equal to one. UCL = c + 3! c LCL = c ! 3" c. Statistics- Set of methods for collecting/analyzing data (the art and science of learning from data). It assesses the average squared difference between data values and the mean. In other words, it is a mathematical discipline to collect, summarize data. Formulas and Tables for AP Statistics. Middle value in ordered data. 2. Start Date – date furthest in the past. In case of Cube, put l = b = h = a, as cube all its sides of equal length, to find the surface area and volumes. Introduction; 9. Cronbach’s alpha coefficient ( ): A coefficient of consistency that measures how well a set of variables or items measures. The COUNT function is used to count the number of cells containing a number. 📌Follow for more Learnbay #probability #conditionalprobability # Jan 27, 2023 · Basic Statistics Formulas: Download Concepts PDF. 0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform. Calculation of mean deviation for ungrouped data involves the following steps : Let us assume the observations x 1, x 2, x 3, …. Descriptive Statistics. Units must be in double quotes. Arrow down to binomial Pdf and press [ENTER]. 🤓The possibility of an event or outcome occurring based on the occurrence of a preceding event or outcome is known as conditional probability. 70 to 0. You can apply these to assess only one variable at a time, in univariate Definitional Formula: SS= ∑ :X− μ ;2 Computational Formula: SS= Q∑X2− :∑X ; 2 R E) Standard Deviation (Ch. Also, we can say that statistics is a branch of applied mathematics. . for example). This chapter discusses these three statistics—mode, median, and mean—and demonstrates the advantages and draw-backs of each using a variety of examples. Statistically Figuring Sample Size 1. Test your knowledge with MCQs on various topics and access smart features like time to answer and smart answer key. FORMULAE 1 Measures of Central Tendency Mean for Ungrouped Data Mean for Grouped Data Weighted Mean Combined Mean n i i1 x X n ¦ In General n i i1 d XA n ¦ k ii i1 k i i1 fx X f ¦ ¦ k ii i1 k i i1 fd X A , f ¦ ¦ k i1 W k i i1 xw w ¦ ¦ 1 1 2 2 12 n X n X X nn k ii i1 k i i1 nX X n ¦ ¦ Median for From the formula, x= + z˙so a z score between 2 and 2 means you are in the interval ( 2˙; + 2˙) and hence by the Empirical Rule 95% of the students taking the exam have z scores between 2 and 2. Statistics is a discipline of mathematics that deals with data analysis and numbers. You will soon understand that statistics and probability work together. Step 2: Find the class mark by taking the midpoint of the upper and lower class limits. 8 Events are subsets of the sample space (A,B,C,). You can use inferential statistics to make estimates and test hypotheses about the whole population of 11th graders in the state based on your sample data. Always remember one thing that it will only count the number. Statistics, on the other hand, is involved with how we handle various data sets using various techniques. While some view it as a standalone scientific field, others classify it as a separate branch within mathematics. Calculates the average of absolute deviations of its argument numbers from their mean. Formula for COUNT function = COUNT(value1, [value2], …) Example of statistical function. List MF20 . If a cell is not blank, it returns FALSE. Each bar represents a group of data and the bars can be compared to each other. For example, the sum of first n terms of a series in sigma notation can be represented as: ∑k=1n Xk ∑ k = 1 n X k. (1) Mean or Average: Mean in theory is defined as the sum of all the elements of a set divided by the number of elements. ) 1. Probability is based entirely on chance. University, Atlanta, GA) The word Oct 14, 2020 · Excel can be used to generate measures of location and variability for a variable. Here are some basic statistics formulas that cover a range of concepts: Mean Formula. Let X be a random variable with mean . This formula is NOT in the function library. AVERAGE. See WARNING below. x n. 1 What is Statistics? In common usage people think of statistics as numerical data—the unem-ployment rate last month, total government expenditure last year, the num-ber of impaired drivers charged during the recent holiday season, the crime-rates of cities, and so forth. We can write the formula for the standard deviation as s = √⅀(𝑥𝑖 − 𝑥̅) 2 𝑛−1 where ⅀ means “the sum of” 𝑥𝑖 represents each value x in the date 𝑥̅ is the mean of the 𝑥𝑖 values Sep 19, 2016 · Process: Formulate the hypothesis. List of Formulae and Statistical Tables . The median is the middle number. 1. Purpose. To collect valid data for statistical analysis, you first need to specify your hypothesesand plan out your research design. Suppose we wish to find. Elsevier, 2013, ISBN 9780124 169715. Description – Graphical and numerical methods for summarizing the data. Jun 28, 2024 · Mean, median, and mode are three important measures of central tendency in statistics. Thus, there are 7 textbooks that have a discount out of 9 books. Data can be represented with a frequency table or with a bar graph. obtain the following formula for the mean ( ̄x). 4. Make a conclusion. Inference – Methods for making predictions about a population (total set of subjects of Jul 17, 2014 · Laboratory Statis tics, Handbook of. Step 3: Tabulate the product of the class mark and its corresponding frequency for each class. In data analysis are primarily used two main statistical methods: descriptive statistics and inferential statistics. Step 3: Find the median by using the following formula: ⎛ n ⎞ ⎜ - F ⎟. The study of data collection, analysis, interpretation, presentation, and organisation is known as statistics. Direct Method to Find Mean. Step 1. 3 days ago · Solved Examples for Class 10 Statistics. Jul 2, 2024 · 102 Most Useful Excel Formulas with Examples Section A – IS FUNCTIONS 1. Modularity rating: 5 Modularity is a strength of this text in both the PDF and interactive online format. Or use the 2-sample version to determine whether two population means differ. Variables are frequently denoted with subscripts, for example, X where the subscript "i" takes on the Repeat this process for formulas that calculate a range of cells (e. x. bf op kz af lk vm or wm qi sz